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SATenstein : automatically building local search SAT solvers from components Khudabukhsh, Ashiqur Rahman 2009

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SATenstein: Automatically Building Local Search SAT Solvers From Components by Ashiqur Rahman KhudaBukhsh A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in The Faculty of Graduate Studies (Computer Science) The University of British Columbia (Vancouver) October 2009 c© Ashiqur Rahman KhudaBukhsh 2009 ii Abstract Designing high-performance solvers for computationally hard problems is a dif- ficult and often time-consuming task. It is often the case that a new solver is created by augmenting an existing algorithm with a mechanism found in a different algorithm or by combining components from different algorithms. In this work, we demonstrate that this task can be automated in the context of stochastic local search (SLS) solvers for the propositional satisfiability problem (SAT). We first introduce a generalized, highly parameterized solver framework, dubbed SATenstein, that includes components drawn from or inspired by ex- isting high-performance SLS algorithms for SAT. In SATenstein, we exposed several design elements in the form of parameters that control both the selection and the behavior of components. We also exposed some parameters that were hard-coded into the implementations of the algorithms we studied. By setting these parameters, SATenstein can be instantiated as a huge number of different solvers, including many known high-performance solvers and trillions of solvers never studied before. We used an automated algorithm configuration procedure to find instantiations of SATenstein that perform well on several well-known, challenging distributions of SAT instances. Overall, we consistently obtained significant improvements over the previous best-performing SLS algorithms, de- spite expending minimal manual effort. iii Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Background and Related Work . . . . . . . . . . . . . . . . . . . 4 2.1 The Satisfiability Problem . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Related Work on Automatic Algorithm Configuration, Portfolios, and Algorithm Synthesis . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.1 Automatic Algorithm Configuration . . . . . . . . . . . . 6 2.2.2 Parameter Tuning and Algorithm Configuration . . . . . 7 2.2.3 Algorithm Selection and Portfolios . . . . . . . . . . . . . 7 2.2.4 Further Related Work . . . . . . . . . . . . . . . . . . . . 9 2.3 Local Search for SAT . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3.1 Different Classes of SLS Algorithms for SAT . . . . . . . 9 2.3.2 Recent Trends in SLS Algorithms . . . . . . . . . . . . . . 14 2.3.3 UBCSAT . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 SATenstein-LS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Implementation and Validation . . . . . . . . . . . . . . . . . . . 22 4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.1 Instance Distributions . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Tuning Scenario and PAR . . . . . . . . . . . . . . . . . . . . . . 26 4.3 Solvers Used for Performance Comparison . . . . . . . . . . . . . 28 4.4 Execution Environment . . . . . . . . . . . . . . . . . . . . . . . 30 Contents iv 5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.1 Comparison with Challengers . . . . . . . . . . . . . . . . . . . . 31 5.2 Comparison with a Hypothetical Oracle . . . . . . . . . . . . . . 36 5.3 Comparison with Complete Solvers . . . . . . . . . . . . . . . . . 36 5.4 Heterogeneous Mixtures of Instance Distributions . . . . . . . . . 39 5.5 Configurations Found . . . . . . . . . . . . . . . . . . . . . . . . 40 6 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . 45 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 A SATenstein-LS User Manual . . . . . . . . . . . . . . . . . . . . . 54 A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 A.2 How to Run SATenstein-LS . . . . . . . . . . . . . . . . . . . . . 54 A.3 Configurable Parameters . . . . . . . . . . . . . . . . . . . . . . . 54 A.4 Example Parameter Configurations . . . . . . . . . . . . . . . . . 55 A.4.1 Novelty+ . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 A.4.2 gNovelty+ . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 A.4.3 SAPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 vList of Tables 2.1 Different search diversification strategies found in Novelty variants. . 12 2.2 Different strategies used in G2WSAT components. . . . . . . . . . . . 14 3.1 SATenstein-LS components. . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Categorical parameters of SATenstein-LS. Unless otherwise mentioned, for more than one parameters, the conditional operator used in the the ‘Active When’ column is AND. . . . . . . . . . . . . . . . . . . . . 19 3.3 Integer parameters of SATenstein-LS. Unless otherwise mentioned, for more than one parameters, the conditional operator used in the the ‘Active When’ column is AND. . . . . . . . . . . . . . . . . . . . . 20 3.4 Continuous parameters of SATenstein-LS. Unless otherwise mentioned, for more than one parameters, the conditional operator used in the the ‘Active When’ column is AND. . . . . . . . . . . . . . . . . . . . . 20 3.5 Design choices for selectFromPromisingList() and corresponding val- ues of selectPromVariable. . . . . . . . . . . . . . . . . . . . . . . 21 3.6 List of heuristics chosen by the parameter heuristic and dependent parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.1 Our six benchmark distributions. . . . . . . . . . . . . . . . . . . . 26 4.2 Integer parameters of SATenstein-LS and the values considered during ParamILS tuning. For each parameter, the known defaults are high- lighted in bold. For parameters that are first introduced in SATenstein-LS, the default values are indicated with an underline. . . . . . . . . . . 27 4.3 Values considered for continuous parameters of SATenstein-LS during ParamILS tuning. For each parameter, the known defaults are high- lighted in bold. For parameters that are first introduced in SATenstein-LS, the default values are indicated with an underline. . . . . . . . . . . 28 4.4 Our eleven challenger algorithms. . . . . . . . . . . . . . . . . . . . 29 4.5 Complete solvers we compared against. . . . . . . . . . . . . . . . . 29 List of Tables vi 5.1 Performance summary of SATenstein-LS and the 11 challengers. Ev- ery algorithm was run 10 times on each instance with a cutoff of 600 CPU seconds per run. Each cell 〈i, j〉 summarizes the test-set perfor- mance of algorithm i on distribution j as a/b/c, where a (top) is the the penalized average runtime; b (middle) is the median of the median runtimes over all instances (not defined if fewer than half of the me- dian runs failed to find a solution within the cutoff time); c (bottom) is the percentage of instances solved (i.e., having median runtime < cutoff). The best-scoring algorithm(s) in each column are indicated in bold, and additionally the best-scoring challenger(s) are indicated with an underline. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.2 Percentage of instances on which SATenstein-LS achieved better me- dian runtime than each of the 11 challengers. Medians were taken over 10 runs on each instance with a cutoff time of 600 CPU seconds per run. When both SATenstein-LS and challenger solved a given instance and achieved indistinguishable median runtimes, we counted that instance as 0.5 for SATenstein-LS and 0.5 for the challenger. . . 34 5.3 Performance summary of SATenstein-LS and the complete solvers. Every complete solver was run once (SATenstein-LS was run 10 times) on each instance with a per-run cutoff of 600 CPU seconds. Each cell 〈i, j〉 summarizes the test-set performance of algorithm i on distribu- tion j as a/b/c, where a (top) is the the penalized average runtime; b (middle) is the median runtime over all instances (for SATenstein-LS, it is the median of the median runtimes over all instances ); c (bottom) is the percentage of instances solved. The best-scoring algorithm(s) in each row are indicated in bold, and additionally the best-scoring com- plete solver(s) are indicated with an underline. An empty cell, 〈i, j〉, means we did not run complete solver i on the test set of distribution j. 38 5.4 Description of heterogenous distributions we used. In each case, we sampled uniformly at random from the component homoge- neous distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.5 Performance summary of SATenstein-LS and the best challenger. Ev- ery algorithm was run 10 times on each instance with a cutoff of 600 CPU seconds per run. Each cell 〈i, j〉 summarizes the test-set penal- ized average runtime (PAR) of algorithm i on distribution j. . . . . . 39 5.6 Performance summary of SATenstein-LS obtained for each category (Handmade, Random, and Industrial) and the Big-mix distribution on a per-distribution basis. Every algorithm was run 10 times on each instance with a cutoff of 600 CPU seconds per run. Each cell 〈i, j〉 summarizes the test-set penalized average runtime (PAR) of algorithm i on distribution j. . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.7 SATenstein-LS configurations. . . . . . . . . . . . . . . . . . . . 40 List of Tables vii 5.8 Distance between SATenstein-LS[D] and each challenger. Each cell 〈i, j〉 summarizes the distance between challenger j and SATenstein-LS[i] as a/b/c, where a (top) is the level 0 distance; b (middle) is the level 1 distance; c (bottom) is the level 2 distance. For each distribution, the closest algorithm distances(s) are highlighted in bold. . . . . . . . . . 43 5.9 SATenstein-LS parameter configuration found for each distribution. . 44 A.1 Name and value mapping SATenstein-LS parameters discussed so far and SATenstein-LS parameters exposed on the command line. . . . . 55 viii List of Figures 3.1 Dependencies between SATenstein-LS parameters. . . . . . . . . . . 18 3.2 Quantile-quantile plots of the run-length distributions for SATenstein-LS vs. UBCSAT (Tompkins and Hoos, 2004) implementations of various well-known SLS solvers. Each implementation was run 5000 times on instance bw large.a with a per-run cutoff time of 600 CPU seconds. The p-value of the performed KS-test is given in the caption of each subfigure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 Quantile-quantile plots of the run-length distributions for SATenstein-LS vs. SAT Competition, 2005 submission of VW (Prestwich, 2005) and SAT Competition, 2007 submission of gNovelty+ (Pham et al., 2008). Each implementation is run 5000 times on benchmark instance bw large.a with a per-run cutoff of 600 CPU seconds. The p-value of the per- formed KS-test is given in the caption of each subfigure. . . . . . . . 24 5.1 Scatter plot of median runtimes of SATenstein-LS[R3SAT] vs PAWS on the test set for R3SAT. Every algorithm was run 10 times on each instance with a cutoff of 600 CPU seconds per run. . . . . . . . . . 34 5.2 Penalized average runtime for our SATenstein-LS solver and the top three challengers for each distribution, as a function of cutoff, based on 10 runs on each test-set instance. . . . . . . . . . . . . . 35 5.3 Scatter plot of median runtimes of SATenstein-LS vs Oracle on the test set for each distribution. . . . . . . . . . . . . . . . . . . . . . . 37 ix Acknowledgements I express my gratitude to my supervisors, Dr. Kevin Leyton-Brown and Dr. Holger H. Hoos. I thank them for giving me exciting work, for always finding time to discuss it with me. In our work and in other projects, they demonstrated how one does good research by doing it themselves. Their help, advice, and example is much appreciated, and has, I believe, enhanced my scholastic abilities considerably. I am grateful to Dr. Jim Little for taking the time to be my second reader. I would like to thank Lin Xu, co-author of our IJCAI 2009 paper. Lin’s guid- ance in designing experiments and generating problem instances were very valu- able. His inputs to my thesis has been very useful. Thanks to Dave Tompkins, who got me started with UBCSAT, a crucial part of working on SATenstein-LS, and Dave Thompson, for his help with my IJCAI presentation. Thanks to Frank Hutter and Chris Fawcett for for their help with ParamILS, as well as for clear- ing up concepts in empirical algorithmics. Thanks also to the participants in the EARG reading-group for many stimulating discussions. I would also like to thank CaraBeth and Hoyt for proofreading my thesis. My stay at UBC was fun-filled and full of extra-curricular activities. I would like to thank all members of our band Algo-Rhythm–Anthony, Baharak, Elan, Emtiyaz, CaraBeth, Lloyd, and Pooja–for so many wonderful moments. I also cherish my experience with a UBC Bengali students’ drama group–Anirban, Angshul, Avik, Dipanjan, Debojit, Madhuja and Sridarshini. Thanks to my friend Lucas and roommate Chaitanya for making the campus life an experience to remember. I would also like to thank Ayan, Imon, Rajat, Sudipto, and Sayantan, my friends from India who were always there when I needed them. My heartfelt thanks go to my manager, Scott M. Roberts, whose encourage- ment made my life at Microsoft a wonderful experience. He was very kind to allow me to attend the IJCAI conference and present this work, knowing that how important it was for me and how hard I worked on this. My acknowledge- ments would not be complete without thanking Jian, my friend and officemate at Microsoft. Finally, I would like to thank my parents, my sisters and close family mem- bers for motivating and supporting me with my studies. xDedication To my cute little friends — Limko, Bolon, and Simba. 1Chapter 1 Introduction In the classic novel Frankenstein; or, The Modern Prometheus by Mary Shelley, a brilliant scientist, Victor Frankenstein was obsessed with the idea of creating a perfect human being by combining scavenged human body parts. Our approach in this work is based on the same general idea: we scavenge components from existing high-performance algorithms for a given problem and combine them to build new high-performance algorithms. Our idea finds support from the fact that many new solvers are created by augmenting an existing algorithm with a mechanism found in a different algorithm (see, e.g., Hoos, 2002; Li et al., 2007b) or by combining components of different algorithms (see, e.g., Pham et al., 2008). Unlike Victor Frankenstein’s creation, our algorithm is built using an automated construction process that enables us to optimize performance for specific types of tasks (i.e., problem instances) with minimal human effort.1 Traditionally, high-performance heuristic algorithms are designed in an it- erative, manual process in which most of the design choices are fixed during development time, leaving only a small number of parameters exposed to the user. In many cases, such unexposed parameters are not clearly visible even at the source code level (e.g., they may be hard-coded as numerical constants). In contrast, we propose a new approach to heuristic algorithm design in which the designer fixes as few design choices as possible at development time, instead ex- posing a huge number of design choices in the form of parameters. This approach removes from the algorithm designer the burden of making early design decisions without knowing how different algorithm components will interact on problem distributions of interest. Instead, it encourages the designer to consider several alternative designs drawn from known solvers as well as novel mechanisms for a given algorithm component. Of course, such flexible, highly parameterized algorithms must be instantiated to solve actual problems. With the availabil- ity of advanced automated parameter configurators and cheap computational resources, for a given distribution, it becomes practical to find good parameter configurations from such huge parameter spaces by using automated parameter configurators (see, e.g., Hutter et al., 2007b; Birattari et al., 2002; Chiarandini et al., 2008) becomes practical. For a more general treatise of this approach, see also Hoos, 2008. Our approach can be seen as a part of the growing trend towards automat- ing the process of constructing high-performance algorithms. Hard combinato- 1This also distinguishes our approach from work by Montes de Oca et al., 2007, which used the Frankenstein metaphor in the context of manually combining algorithm components drawn from existing high-performance particle swarm optimisation procedures. Chapter 1. Introduction 2 rial problems arise in numerous real-life applications. The demand for efficient solvers for such problems is ever increasing. With computational resources get- ting cheaper day by day, it becomes possible to reduce human effort by utilizing cheap computing power to automate the algorithm-building process. Examples of work along these lines are discussed in Chapter 2. Although our general idea is not specifically tailored to a particular do- main, in this work we address the challenge of constructing stochastic local search (SLS) algorithms for the propositional satisfiability problem (SAT): an NP-complete problem of great interest for scientists and industry people alike. Improving the state of the art in this domain is not an easy task. Indeed, a substantial amount of research and engineering effort has been expended in building SLS algorithms for SAT since the late 1980s (see, e.g., Selman et al., 1992; Hoos, 2002; Pham et al., 2008). SLS-based solvers show a continuing trend of dominance over other classes of SAT solvers on random satisfiable instances. Since 2003, they won all medals in the random satisfiable category of the SAT Competition (SAT Competition, 2009). The SATzilla solvers that won the SAT+UNSAT tracks of two of the three categories (handmade and random) in the 2007 SAT Competition also make use of SLS procedures. Overall, three new SLS-based solvers emerged as winners in the random satisfiable category and new SLS-based solvers have been introduced roughly every year for the past 16 years. This highlights the research community’s continuing interest in this area and indicates the scope for further performance improvement. In some cases, the performance improvements achieved by these solvers are known to be the result of a design process during which a large number of candidate designs were tested (see, e.g., McAllester et al., 1997); indeed, it is our belief that this is usually the case. We have leveraged this rich body of work on SLS algorithms for SAT to design our generalized algorithm SATenstein-LS (KhudaBukhsh et al., 2009). This algorithm draws mechanisms from two dozen existing high-performance SLS SAT solvers and also incorporates many novel strategies. Our design space contains a total of 4.82 × 1012 candidate solvers, including both most of the state-of-the-art SLS SAT solvers known from the literature and a vast number of novel designs that have not been studied before. Another important element is the algorithm configurator we used. In princi- ple, any powerful algorithm configuration tool could be used to find SATenstein configurations that perform well on problem distributions of interest. In our case, we used an off-the-shelf algorithm configuration procedure, ParamILS (Hutter et al., 2007b, 2008). ParamILS has been successfully used to configure a diverse range of highly parameterized applications (see, e.g., CPLEX (Hutter et al., 2008); Spear (Hutter et al., 2007a); Chiarandini et al., 2008). We configured our generalized design, SATenstein-LS, on six well-known SAT distributions ranging from hard random 3-SAT instances to SAT-encoded factoring and software verification problems. We evaluated the performance of the new solvers thus obtained by comparing them against 11 state-of-the-art SLS-based solvers and 2 state-of-the-art complete solvers. Our results show that our new solvers outperformed existing high-performance SLS solvers in Chapter 1. Introduction 3 all six distributions. Among these six distributions, our new solvers improved the state of the art for four. On the remaining two distributions — in par- ticular the SAT-encoded software verification instances — our new solvers still emerged as the best-performing SLS-based solvers and considerably narrowed the performance gap between SLS-based algorithms and DPLL-based solvers. We analyzed the configurations of these new solvers and related them to exist- ing high-performance solvers in various ways. We defined a distance measure between two SATenstein-LS configurations and computed the distance between our new solvers and known high-performance SLS solvers. Based on this mea- sure, we found that many of our new solvers were considerably different from existing high-performance solvers. The remainder of this thesis is organized as follows. After a discussion of related work (Chapter 2), we describe the design and implementation of SATenstein-LS (Chapter 3). This is followed by a description of the setup we used for empirically evaluating SATenstein-LS (Chapter 4) and a presentation and discussion of the results from our experiments (Chapter 5). We end with some general conclusions and an outlook on possible future work (Chapter 6). 4Chapter 2 Background and Related Work In this chapter, we first formally define the propositional satisfiability problem (SAT) and discuss why SAT is an important problem domain. We also mention the state of the art techniques used in SAT solving. Next, we discuss various lines of research that are related to our work on SATenstein. We then highlight some important aspects of four broad categories of stochastic local search (SLS) SAT solvers. We finally conclude with outlining recent research trends in SLS algorithm development. 2.1 The Satisfiability Problem The propositional satisfiability problem (SAT) asks, for a given propositional formula F, whether there exists a complete assignment of truth values to the variables of F under which F evaluates to true (see, e.g., Biere et al., 2009). Such complete assignments are also known as models. F is called satisfiable if there exists at least one such model. Otherwise, the formula is called unsatisfi- able. A propositional satisfiability problem instance is usually represented as a conjunction of disjunctions, where each disjunction has one or more arguments that may either be variables or the negation of variables. Such arguments are known as literals. Each disjunction is called a clause. This syntactic form of representing a propositional satisfiability problem instance is known as conjun- tive normal form. Thus the goal for a SAT solver is to find a variable assignment that satisfies all clauses or to prove that no such assignment exists. Many of the hard combinatorial problems that arise in practical scenarios be- long to the complexity class of so-called NP-complete problems (see, e.g., Cook, 1971; Applegate et al., 2006; Biere et al., 1999; Fraenkel, 1993; Pop et al., 2002). It is widely believed that there does not exist any deterministic, polynomial time solution to any of these NP-complete problems. SAT is one of the most-studied NP-complete problems. In fact, there are conferences1 and journals2 solely dedicated to the satisfiability problem. SAT is also important because NP- complete problems from other classes can be encoded as SAT problem instances 1For example: Proceedings of the International Conferences on Theory and Applications of Satisfiability Testing. 2For example: Journal on Satisfiability, Boolean Modeling and Computation (see, http: //www.isa.ewi.tudelft.nl/Jsat/). Chapter 2. Background and Related Work 5 and effectively solved by using a SAT solver. For example, using this approach, problems arising in applications such as planning (e.g., Kautz and Selman, 1996, 1999), scheduling (Crawford and Baker, 1994), graph-coloring (Gelder, 2002), bounded model checking (Biere et al., 1999), and formal verification (Stephan et al., 1992) have been effectively solved. Over the last few decades, a considerable amount of research and engineer- ing effort has been expended to improve the state of the art in SAT solving. Present high-performance SAT solvers include tree-search algorithms (see, e.g., Sörensson and Eén, 2007; Heule and Maaren, 2007a; Biere, 2008; Dubois and Dequen, 2001), local search algorithms (see, e.g., Hutter et al., 2002; Hoos, 2002; Pham and Anbulagan, 2007; Li and Huang, 2005; Prestwich, 2005; Pham et al., 2008; Li et al., 2007b) and resolution-based preprocessors (see, e.g., Davis and Putnam, 1960; Subbarayan and Pradhan, 2005; Dechter and Rish, 1994; Bac- chus and Winter, 2003; Bacchus, 2002a,b). Every year, competitions are held (namely the SAT Competition and the SAT Race) where new state-of-the-art solvers emerge. The trend of continuing performance improvement in the solvers that take part in SAT competitions suggests that there is scope for yet further performance improvement. 2.2 Related Work on Automatic Algorithm Configuration, Portfolios, and Algorithm Synthesis There is a large body of literature in AI and related areas that deals with automated methods for building effective algorithms. This includes work on automatic algorithm configuration (see, e.g., Gratch and Dejong, 1992; Minton, 1993), algorithm selection (see, e.g., Guerri and Milano, 2004; Carchrae and Beck, 2005; Xu et al., 2008), algorithm portfolios (see, e.g., Gomes and Selman, 2001; Gagliolo and Schmidhuber, 2006), and, to some extent, genetic program- ming (see, e.g., Fukunaga, 2002, 2004; Oltean, 2005) and algorithm synthesis (see, e.g., Westfold and Smith, 2001; Monette et al., 2009; Gaspero and Schaerf, 2007). The goal of minimizing human effort in algorithm design and implemen- tation addresses different areas of human involvement and tries to achieve its goal in different ways. On one hand, there is a line of work on automatic al- gorithm configuration aimed at coming up with new high-performance solvers, often custom-tailored for a given problem distribution. On the other hand, there is a line of research on algorithm portfolios and algorithm selection that tries to make the most out of existing solvers without creating a new solver. For a more detailed discussion of automated algorithm design, see also Hoos, 2008. In broad strokes, our approach can be differentiated from existing work in three key ways. First, we keep the two phases in our approach completely separate and independent from each other. In the first phase, we specify the combinatorial design space. In the second, we search the space for a good pa- rameter configuration. Keeping these two phases separate allows us to enrich Chapter 2. Background and Related Work 6 the design space by adding newer design alternatives without worrying how that would affect the second phase of our approach. Also, we can plug in any ad- vanced configurator without affecting the specified combinatorial design space. Second, we use an automated algorithm configuration tool for searching the de- sign space. Every time we get a new problem distribution, this tool can be used to find a new algorithm configuration. This minimizes human involvement in the algorithm design process, since otherwise a considerable amount of human development time can be spent in re-engineering an old algorithm to make it per- form well on a new distribution. Third, experimental results show that we have been able to improve the state of the art in four of the six widely studied prob- lem distributions of propositional satisfiability problem (SAT) (KhudaBukhsh et al., 2009). Detailed discussion of our results are presented in Chapter 5. 2.2.1 Automatic Algorithm Configuration In the past, a lot of interesting work has been done on automatic algorithm configuration. For example, Minton (1993) used meta-level theories to produce distribution-specific versions of generic heuristics, and then found the most use- ful combination of these heuristics by evaluating their performance on a small set of test instances. Our work is different from Minton’s approach in the follow- ing ways. First, instead of producing distribution-specific versions of candidate heuristics for each distribution, we lay out a generalized, highly parameterized framework, which remains the same for every distribution, by exposing design choices as parameters. Second, our work is very different in terms of scale; we explore a space of 4.82× 1012 candidate solvers, whereas Minton considered at most 100 possible heuristics. Finally, our results indicate that we improved the state of the art for most of the distributions we considered, whereas in work by Minton, (1993), the performance of the resulting algorithms was comparable with that of algorithms designed by a skilled programmer, but not an algorithm expert. Gratch and Dejong (1992) presented a system that starts with a STRIPS- like planner, and augments it by incrementally adding search control rules. In SATenstein, we do not augment an existing solver; rather, our goal is to design a method for automatically building new solvers by combining components from various existing solvers. Fukunaga’s (2002; 2004) genetic programming approach, CLASS (Composite heuristic Learning Algorithm for SAT Search), is relevant to our own work in that it is motivated by a similar goal: the automated construction of local search heuristics for SAT. His approach is called In this approach, the con- sidered design space is potentially unbounded, whereas in our approach, we explore a finite design space of 4.82 × 1012 candidate solvers. In order to con- struct variable selection mechanisms for a generic SLS algorithm, Fukunaga only considered GSAT-based and WalkSAT-based SLS algorithms developed up to the year 2000. Additionally, candidate variable selection mechanisms were eval- uated mostly on Random-3-SAT instances and graph coloring instances with at most 250 variables. In contrast, we considered several mechanisms, including Chapter 2. Background and Related Work 7 variable selection in SLS algorithms up to the year 2007. Our target problems were much bigger, as they considered SAT instances with up to 998 variables and 9623 clauses, and also included instances that are hard for local search al- gorithms, such as software verification problems (Clarke et al., 2004) and SAT encoded factoring problems (Uchida and Watanabe, 1999). The two steps in our approach - the specification of possible design choices and the automatic construction of algorithms for a given distribution - are kept separate from each other. The off-the-shelf configurator we used can be easily replaced by any other algorithm configurator without affecting our first step. On the other hand, the genetic programming procedure used by Fukunaga is tailored specifically to searching the design space considered in his study. Finally, while Fukunaga’s approach could in principle be used to obtain high-performance solvers for spe- cific types of SAT instances, to our knowledge this potential was never realized. The best automatically-constructed solvers obtained by Fukunaga only achieved a performance level similar to that of the best WalkSAT variants available in 2000 on moderately-sized SAT instances. In contrast, our new SATenstein-LS solvers performed substantially better than current state-of-the-art SLS-based SAT solvers on a broad range of challenging, modern SAT instances. 2.2.2 Parameter Tuning and Algorithm Configuration In order to allow flexibility and to give good performance over a broad range of instances, many recent high-performance algorithms expose configurable param- eters (see, e.g., CPLEX, SAPS (Hutter et al., 2002)). Finding a good parameter configuration is often the most critical task for obtaining good performance on a given problem instance or problem distribution. There are several techniques found in the literature that attempts to solve the problem of algorithm config- uration. The two most prominent examples are F-race (Birattari et al., 2002; Balaprakash et al., 2007) and ParamILS (Hutter et al., 2007b; Hutter et al.). F-race is a racing algorithm that is used for designing meta-heuristics. In this approach, a non-parametric statistical test is used to iteratively filter out config- urations that are significantly worse than others. Such filtering continues until a cutoff time is reached and only a small number of good configurations are left. ParamILS (Hutter et al., 2007b; Hutter et al.) is a parameter configura- tion tool based on iterated local search (ILS). ParamILS has been successfully used in configuring a diverse set of applications (see, e.g., Hutter et al., 2007a; Chiarandini et al., 2008). The main disadvantage of F-Race over ParamILS is its low scalability in comparison with ParamILS. Since SATenstein-LS has a vast design space, we chose ParamILS as our automated algorithm configurator. 2.2.3 Algorithm Selection and Portfolios There are several other techniques aimed at making the most out of the exist- ing high-performance solvers. Examples include algorithm selection and algo- rithm portfolios. For a given problem instance or problem distribution, often we have to solve an “algorithm selection problem” (Rice, 1976): which algorithm(s) Chapter 2. Background and Related Work 8 should be run in order to minimize some performance objective, such as expected runtime? Different machine learning techniques are applied to solve the algo- rithm selection problem (see, e.g., Leyton-Brown et al., 2002; Guerri and Milano, 2004; Carchrae and Beck, 2005; Xu et al., 2008). In Leyton-Brown et al., (2002), statistical regression techniques are applied to identify the connection between instance-related features and its predicted hardness. In a similar vein, Guerri and Milano (2004) used a decision tree-based technique to identify the connec- tion between the structure of a problem instance and algorithm performance. Although our problem can be reformulated as an algorithm selection problem, it is obvious that such techniques are infeasible for selecting from a set of 4.82×1012 candidate solvers. However, there exists a conceptual relationship between our work and algorithm selection approaches. For example, SATzilla (Nudelman et al., 2004a; Xu et al., 2008) performed per-instance algorithm selection based on the algorithms’ predicted performance. Both SATzilla and SATenstein-LS leverage existing SAT solvers to achieve better performance (also, both are named after movie monsters!). However, SATzilla and SATenstein-LS are dif- ferent in the following ways. First, SATzilla chooses from a relatively small set of known algorithms, which includes non-SLS-based algorithms. In contrast, SATenstein-LS can instantiate trillions of SLS solvers, including a vast number of solvers never studied before. Second, in order to perform algorithm selection, SATzilla uses empirical hardness models (see, e.g., Leyton-Brown et al., 2002, 2009; Nudelman et al., 2004b) to predict runtimes for each algorithm in the portfolio on a given SAT instance, based on instance-related features. Using this approach, creating such performance-predicting models for SATenstein-LS is infeasible, and by design, SATenstein-LS should be used on a per-distribution basis. On the other hand, the approaches for SATzilla and SATenstein-LS can be seen as complementary, as SATenstein-LS solvers can be used in a SATzilla portfolio. Indeed, the latest version of SATzilla, SATzilla2009 R, which per- formed extremely well in the random category in the 2009 SAT Competition (winning a gold medal in the random SAT+UNSAT category and placing 4th in the random SAT category), makes use of several solvers constructed using the SATenstein-LS framework (Xu et al., 2009). It has often been observed that a significant improvement in performance can be obtained by running multiple algorithms in parallel. Also, for stochastic algorithms, running multiple copies of the same algorithm can be beneficial. Al- gorithm portfolios aim to achieve performance improvement by running multiple algorithms concurrently (Gomes and Selman, 2001). In general, all or a subset of the algorithms in a portfolio are allocated CPU shares in a multi-tasking environment or run concurrently on a multi-processor (or multi-core) comput- ing platform. Extensions of this general idea include the dynamic reallocation of CPU share between the component algorithms (Gagliolo and Schmidhuber, 2006) and communication between component algorithms (Huberman et al., 1997). Chapter 2. Background and Related Work 9 2.2.4 Further Related Work The Frankenstein metaphor of algorithm design has also been used in work by Montes de Oca et al., 2007, where a Particle Swarm Optimization (PSO) algorithm is created by combining algorithm components drawn from existing high-performance PSO algorithms. In this approach, specific designs for each of the components are hand-picked by the algorithm designer. In contrast, we specify a combinatorial design space from which we use an automated al- gorithm configurator to find a good design for a given problem distribution. Clearly, Frankenstein’s PSO can be seen an example of manual algorithm de- sign, whereas, our goal is to automate the algorithm-building process. Existing work on algorithm synthesis is mostly focused on automatically generating algorithms that satisfy a given formal specification or that solve a specific problem from a large and diverse domain (see, e.g., Westfold and Smith, 2001; Monette et al., 2009; Gaspero and Schaerf, 2007). In contrast, our work is focused on finding an efficient solver from amongst a huge space of candidate solvers that are all guaranteed to be correct by construction. Clearly, these two approaches are focused on solving different types of problems. 2.3 Local Search for SAT The state-of-the-art algorithms for solving certain classes of SAT instances in- clude Stochastic Local Search (SLS) algorithms. For a given SAT problem in- stance, the objective function, which an SLS SAT solver attempts to minimize, is the number of unsatisfied clauses. In most cases, the score of a variable x with respect to an assignment A, scoreA(x ), is defined as the decrease of the value of the objective function when x is flipped (flipping a variable means changing its truth value). A typical SLS algorithm for SAT consists of an initialization phase and a search phase. In the initialization phase, a complete assignment of truth values to all variables is given. At each step in the search phase, the truth value of a heuristically-determined variable is changed. Since each step essentially means flipping a variable, the terms flip and step are often used in- terchangeably. Sometimes, we may have a null flip where no variable is flipped. At each search step, the set of variables that are considered for flipping is called the neighborhood. The search process is terminated when either a satisfying assignment is found or a given bound on the runtime or run-length is reached or exceeded. Almost all SLS algorithms for SAT are by nature incomplete, i.e., they cannot prove unsatisfiability. 2.3.1 Different Classes of SLS Algorithms for SAT Existing SLS-based SAT solvers can be grouped into four broad categories: GSAT-based algorithms (Selman et al., 1992), WalkSAT-based algorithms (Sel- man et al., 1994), dynamic local search algorithms (Hutter et al., 2002; Thorn- ton et al., 2004), and G2WSAT variants (Li and Huang, 2005). The following subsections describe the major features of each of these categories. Chapter 2. Background and Related Work 10 Procedure GSAT(CNF formula F,maxTries,maxSteps) Input: CNF formula F, positive integers maxTries and MaxSteps Output: model of F or ‘no solution found’ for try := 1 to maxTries do a:= randomly chosen assignment of the variable in formula F ; for step :=1 to maxSteps do if a satisfies F then return a ; x := randomly selected variable flipping that minimizes the number of unsatisfied clause ; a := a with x flipped; return ‘no solution found’ GSAT GSAT (Selman et al., 1992) was one of the earliest SLS SAT solvers. As shown in Procedure GSAT, at each step, GSAT computes the score of each variable using a scoring function, then flips the variable with the best score.3 To describe the scoring function of GSAT, we need to define the MakeCount and BreakCount of a variable. The MakeCount of a variable with respect to an assignment A is the number of previously-unsatisfied clauses that will be satisfied if the variable is flipped. Similarly, the BreakCount of a variable with respect to an assignment A is the number of previously-satisfied clauses that will be unsatisfied if the variable is flipped. The scoring function of GSAT is MakeCount - BreakCount. Variants of GSAT introduced many techniques that were later used by other SLS solvers. For example, GWSAT (Selman and Kautz, 1993) performs a conflict- directed random walk step with probability wp. Otherwise, it performs a regular GSAT step. Conflict-directed random walk is an example of a search diversifica- tion strategy that was later used by many SLS solvers. In an SLS search step, for a given scoring function, multiple variables can have the same score. GSAT breaks such ties randomly. HSAT (Gent and Walsh, 1993) introduced a new tie- breaking scheme in which ties are broken in favor of the least-recently-flipped variable. In subsequent SLS solvers, breaking ties randomly and breaking in the favor of the least-recently-flipped variable were prominent tie-breaking schemes. GSAT now has only historical importance, as there is a substantial performance gap between GSAT and recent state-of-the-art SLS solvers. WalkSAT Architecture The major difference between WalkSAT algorithms and GSAT algorithms is the neighborhood they consider. For GSAT algorithms, the neighborhood is the full set of variables in the formula. For a WalkSAT algorithm, the neighborhood consists of variables belonging to a selected unsatisfied clause. As shown in 3The pseudocodes of GSAT and WalkSAT are taken from Hoos and Stützle, 2005. All our pseudocodes follow the same notations as in Hoos and Stützle, 2005. Chapter 2. Background and Related Work 11 Procedure WalkSAT(CNF formula F,maxTries,maxSteps) Input: CNF formula F, positive integers maxTries and MaxSteps, heuristic function slc Output: model of F or ‘no solution found’ for try := 1 to maxTries do a:= randomly chosen assignment of the variable in formula F ; for step :=1 to maxSteps do if a satisfies F then return a ; c := randomly selected clause unsatisfied under a ; x := variable selected from c according to heuristic function slc; a := a with x flipped; return ‘no solution found’ Procedure WalkSAT, at each search step, a WalkSAT algorithm picks an unsat- isfied clause (the most popular way is picking an unsatisfied clause uniformly at random), and flips a variable (depending on some heuristic) from that clause. WalkSAT/SKC (Selman et al., 1994) was one of the earliest algorithms belonging to WalkSAT architecture. A notable difference between WalkSAT/SKC and many other SLS algorithms, including GSAT, is that WalkSAT/SKC uses BreakCount as its scoring function. Novelty (McAllester et al., 1997) and its several variants are among the most prominent WalkSAT algorithms. Novelty uses the same scoring function as GSAT and also takes the age of a variable into account (the number of flips performed since it was last flipped). More precisely, Novelty picks a random unsatisfied clause and computes the variables with highest and second-highest scores. Ties are broken in favor of the least-recently-flipped variable. If the vari- able with the highest score is not the most-recently-flipped variable within the clause, then it is deterministically selected for flipping. Otherwise, it is selected with probability (1 - p) where p is a parameter called the noise setting (with probability p, the second-best variable is selected). Novelty represents an inter- esting advancement in the research of SLS solvers. However, it often gets stuck in local minima of the objective functions since the selection is restricted only to the two best variables. To overcome this problem, Novelty was augmented with a probabilistic conflict-directed random walk (Hoos, 1999). The resulting algorithm, Novelty+, showed much better performance than Novelty on many of the benchmark instances. Table 2.1 summarizes different search diversifica- tion strategies found in Novelty variants. Other variants include R-Novelty, R-Novelty+, Novelty+p (Novelty with lookahead), and RANOV (Pham and An- bulagan, 2007) (Novelty+ with a preprocessor). The performance of the new algorithm Novelty+, although more robust than Novelty, still depends a lot on a suitable noise setting. In adaptNovelty+ (Hoos, 2002), a reactive mechanism that adaptively changes the noise parameter was first introduced. This reactive mechanism is extended to many SLS solvers (see, Chapter 2. Background and Related Work 12 Novelty variant Search diversification strategy Novelty+ (Hoos, 1999) flips a randomly selected variable Novelty++ (Li and Huang, 2005) flips the least recently flipped variable Novelty++′ (Li et al., 2007a) randomly flips a variable excluding the best and the second-best variable Table 2.1: Different search diversification strategies found in Novelty variants. Procedure Dynamic Local Search(CNF formula F, maxTries, maxSteps) Input: CNF formula F, positive integers maxTries and MaxSteps Output: model of F or ’no solution found’ for try := 1 to maxTries do a:= randomly chosen assignment of the variable in formula F ; initialize clause-weight for each clause ; for step :=1 to maxSteps do if a satisfies F then return a ; x := variable selected using a scoring function that considers clause-weights; a := a with x flipped; update clause-weight for each clause ; return ‘no solution found’ e.g., Li et al., 2007b) yielding improved performance. The idea of considering flip history is exploited in various ways in different SLS solvers. These include considering the age of a variable (e.g., in Novelty), tabu search (e.g., in WalkSAT/TABU (McAllester et al., 1997)) and flipping the least-recently-flipped variable (e.g., in Novelty++). VW (Prestwich, 2005), one of the most recent WalkSAT algorithms, considers flip counts (the total number of times a variable is flipped). Dynamic Local Search Algorithms The most prominent feature of dynamic local search (DLS) algorithms is the use of “clause penalties” or “clause weights”. The general structure of a DLS algorithm is presented in Figure 3. At each step, the clause penalty of an unsat- isfied clause is increased (this increase can be additive (Thornton et al., 2004) or multiplicative (Hutter et al., 2002)). In this way, information about the dif- ficulty of solving a given clause is recorded in its associated clause penalty. In order to prevent an unbounded increase in weights and to keep the informa- tion about the difficulty of solving a given clause recent, occasional smoothing steps are performed to reduce them. The scoring function is the sum of the clause penalties of all unsatisfied clauses. For prominent DLS solvers like SAPS, Chapter 2. Background and Related Work 13 Procedure G2WSAT(CNF formula F, maxTries, maxSteps) Input: CNF formula F, positive integers maxTries and MaxSteps, deterministic heuristic function dhf and Novelty variant Output: model of F or ’no solution found’ for try := 1 to maxTries do a:= randomly chosen assignment of the variable in formula F ; put decreasing variable(s) in promising list ; for step :=1 to maxSteps do if a satisfies F then return a ; if promising list is empty then c := randomly selected clause unsatisfied under a ; x := variable selected from c according to Novelty variant; else x := variable selected from promising list according to dhf ; a := a with x flipped ; update promising list ; return ‘no solution found’ RSAPS (Hutter et al., 2002), and PAWS (Thornton et al., 2004), the neighborhood consists of variables that appear in at least one unsatisfied clause. G2WSAT Variants G2WSAT (Li and Huang, 2005) can be viewed as a combination of the GSAT and WalkSAT architectures. Similar to GSAT, G2WSAT has a deterministic greedy component that looks at a large number of variables belonging to a promising list data structure that contains promising decreasing variables. As shown in Procedure G2WSAT, if the list has at least one promising decreasing variable, G2WSAT deterministically selects the variable with the best score for flipping. Ties are broken in favor of the least-recently-flipped variable. If the list is empty, G2WSAT executes its stochastic component, a Novelty variant that belongs to the WalkSAT architecture. The definition of a promising decreasing variable is somewhat technical; skip- ping this would not affect understanding the rest of the content in this thesis. However, we present the definition for the sake of completeness. A variable x is said to be decreasing with respect to an assignment A if scoreA(x ) > 0 (the scoring function is MakeCount - BreakCount). A promising decreasing variable is defined as follows: 1. For the initial random assignment A, all decreasing variables with respect to A are promising. 2. Let x and y be two different variables where x is not decreasing with respect to A. If, after y is flipped, x becomes decreasing with respect to Chapter 2. Background and Related Work 14 the new assignment, then x is a promising decreasing variable with respect to the new assignment. 3. As long as a promising decreasing variable is decreasing, it remains promis- ing with respect to subsequent assignments in local search. . Table 2.2 summarizes the different ways the two components of G2WSAT can vary. Apart from G2WSAT (Li and Huang, 2005), all G2WSAT variants use the reactive mechanism found in adaptNovelty+ (Hoos, 1999). gNovelty+ (Pham et al., 2008), the winner of 2007 SAT Competition in the random satisfiable category also uses clause penalties and smoothing found in dynamic local search algorithms (Thornton et al., 2004). G2WSAT component Different strategies Heuristic selection Novelty+ (Li et al., 2007b) Novelty++ (Li and Huang, 2005) Novelty++′ (Li and Huang, 2005) Novelty+p (Li et al., 2007b) Variable selection from promising list Select the variable with highest score, breaking ties in favor of the least-recently-flipped variable (Li and Huang, 2005). Select the least recently flipped variable (Li et al., 2007b). Table 2.2: Different strategies used in G2WSAT components. 2.3.2 Recent Trends in SLS Algorithms Recent developments on SLS algorithms show an increasing trend towards cre- ating new algorithms by combining components of different algorithms, albeit manually. For example, the augmentation of an existing algorithm with mecha- nisms from another’s can be observed in adaptG2WSAT (Li et al., 2007b). There, the adaptive noise mechanism, first introduced in Novelty+ (Hoos, 2002), was successfully combined with an existing algorithm G2WSAT (Li and Huang, 2005), resulting in an improved algorithm. gNovelty+ (Pham et al., 2008), the most re- cent SAT Competition winner in the random satisfiable category, can be viewed as a manually-designed algorithm that combines components from G2WSAT and Ranov (Pham and Anbulagan, 2007). The design of gNovelty+ was largely guided by the fact that G2WSAT and Ranov tend to do well on different types of random-k-SAT instances. The recent high-performance algorithm Hybrid (Wei et al., 2008) combines the strengths of two different algorithms: VW (Prestwich, 2005) and adaptG2WSATp (Li et al., 2007b). Such successes indicate that the per- formance of SLS solvers can be improved by combining algorithm components. Chapter 2. Background and Related Work 15 Without an automated approach, we believe that this type of design will not be able to reach its full potential, since there can be many possible combinations, and exhaustively searching all of them is typically infeasible. Also, manual de- sign is largely guided by an algorithm practitioner’s expertise and intuition. Relying on such intuition may lead to the neglect of unintuitive combinations that perform well in practice. 2.3.3 UBCSAT UBCSAT (Tompkins and Hoos, 2004) is an SLS solver implementation and ex- perimentation environment for SAT. It has already been used to implement many existing high-performance SLS algorithms from the literature. (e.g., SAPS Hutter et al., 2002, adaptG2WSAT+ Li et al., 2007b). These implementations generally match or exceed the efficiency of implementations by the original au- thors. UBCSAT implementations have therefore been widely used as reference implementations (see, e.g., Prestwich, 2005; Kroc et al., 2009) for many well- known local search algorithms. UBCSAT provides a rich interface that includes numerous statistical and reporting features facilitating empirical analysis of SLS algorithms. Many existing SLS algorithms for SAT share common components and data structures. The general design of UBCSAT allows for the reuse and extension of such common components and mechanisms. This made UBCSAT a suitable environment for the implementation of SATenstein-LS. All the algorithm com- ponents and mechanisms used in SATenstein-LS were already implemented in UBCSAT.4 However, many of these components were never used within a single algorithm and were optimized in a way that there coexistence inside a single algorithm might lead to erroneous behavior. Because of this, creating SATenstein-LS was a major software development task that required thorough validation of each of the underlying mechanisms and components used. At a conceptual level, UBCSAT and SATenstein-LS are quite different. UBCSAT has a set of well-known solvers implemented in it in a stand-alone fashion; UBCSAT does not create new solvers by combining these solvers’ components. SATenstein-LS, on the other hand, is a highly parameterized SLS solver that draws components from several well-known SLS algorithms. SATenstein-LS can be configured as trillions of SLS solvers that have never been studied be- fore. In combination with an automatic algorithm configurator, SATenstein-LS is indeed a framework for automatically finding new SLS solvers. 4During the implementation phase of SATenstein-LS, we included several recent high- performance algorithms in the UBSAT implementation suite (e.g., VW, G2WSAT, adaptG2WSAT+, and adaptG2WSATp). In UBCSAT versions prior to 1.1, many of the components used in these algorithms were not implemented. 16 Chapter 3 SATenstein-LS In this chapter, we describe the design of SATenstein-LS, a highly parameter- ized, stochastic local search (SLS) SAT solver. SATenstein-LS draws compo- nents from several high-performance SLS SAT solvers as well as uses various novel mechanisms. We first present a high-level pseudocode of SATenstein-LS and explain the functionalities of different building blocks used in our design. We then present all the parameters that we expose to the end user to configure SATenstein-LS. We conclude with the results of our experiments that we con- ducted to confirm that different mechanisms of SLS solvers have been correctly implemented in SATenstein-LS. 3.1 Design As mentioned already, almost all SLS algorithms for SAT can be categorized into four broad categories: GSAT-based algorithms, WalkSAT-based algorithms, dy- namic local search algorithms and G2WSAT variants. Since none of the state- of-the-art SLS solvers is GSAT-based, SATenstein-LS has been constructed by drawing its components from algorithms belonging to the three remaining broad categories. Procedure SATenstein-LS presents very high-level pseudocode of SATenstein-LS. The pseudocode is divided into five building blocks, B1–B5. Any SATenstein-LS instantiation has the following general structure: 1. Optionally execute B1, which performs search diversification. 2. Execute either B2, B3 or B4, which represent G2WSAT-based, WalkSAT- based and dynamic local search algorithms, respectively. 3. Optionally execute B5, to update data structures such as promising list, clause penalties, dynamically adaptable parameters or tabu at- tributes. Each of our building blocks is composed of one or more components (listed in Table 3.1). Some of the components are shared across different building blocks. Each component is configurable by one or more parameters. Out of 41 total parameters, SATenstein-LS has 16 continuous parameters (listed in Table 3.4), 19 categorical parameters (listed in Table 3.2), and 6 integer parameters (listed in Table 3.3). All these parameters are exposed on the command line and tuned by using our automatic configurator. Many of these parameters conditionally depend on other parameters. A high-level representation of such dependencies is Chapter 3. SATenstein-LS 17 Procedure SATenstein-LS(CNF formula φ) Input: CNF formula φ; real number cutoff ; booleans performDiversification, singleClauseAsNeighbor, usePromisingList Output: Satisfying variable assignment Start with random Assignment A; Initialise parameters; while runtime < cutoff do if A satisfies φ then return A; VarFlipped ← FALSE; if performDiversification then B1 if within probability diversificationProbability() then B1 Execute B1; if Not VarFlipped then if usePromisingList then B2 if promisingList is nonempty then B2 y ← selectFromPromisingList() ; else B2 c ← selectClause(); B2 y ← selectHeuristic(c) ; else if singleClauseAsNeighbor then B3 c ← selectClause(); B3 y ← selectHeuristic(c) ; else B4 sety ← selectSet(); B4 y ← tieBreaking(sety); flip y ; B5 update(); shown in Figure 3.1. The total number of valid SATenstein-LS instantiations is about five trillions. This of course depends on the number of discrete values we consider for each of our continuous parameters (see Table 4.3) and the number of values we consider for each of our integer parameters (see Table 4.2). With a finer grid, the number of valid SATenstein-LS instantiations will increase. We now give a high-level description of each of the building blocks. B1 is constructed using the SelectClause(), DiversificationStrategy() and Diversifica- tionProbability() components. SelectClause() is configured by one categorical parameter and, depending on its value, either selects an unsatisfied clause uni- formly at random or selects a clause with probability proportional to its clause penalty (Tompkins and Hoos, 2004). Component diversificationStrategy() can be configured by a categorical parameter to do any of the following with proba- bility diversificationProbability(): flip the least recently flipped variable (Li and Huang, 2005), flip the least frequently flipped variable (Prestwich, 2005), flip the variable with minimum variable weight (Prestwich, 2005), or flip a randomly selected variable (Hoos, 2002). Block B2 instantiates G2WSAT-based algorithms that use a data structure promising list (for detailed definition, see Chapter 2.3.1) that keeps track of a set of variables considered for being flipped. In the literature on G2WSAT, Chapter 3. SATenstein-LS 18 Component Block params poss. values Based on diversificationStrategy() 1 1 4 Pham et al., 2008 Hoos, 2002 Prestwich, 2005 Li and Huang, 2005 SelectClause() 1, 2, 3 1 2 Tompkins and Hoos (2004) diversificationProbability() 1 4 24 Pham et al., 2008 Hoos, 2002 Prestwich, 2005 Li and Huang, 2005 selectFromPromisingList() 2 4 ≥ 4341 Li and Huang, 2005 Li et al., 2007b Li et al., 2007a Pham et al., 2008 selectHeuristic() 2, 3 9 ≥ 1220800 Hoos, 2002 Li and Huang, 2005 Li et al., 2007b Li et al., 2007a Prestwich, 2005 Selman et al., 1994 McAllester et al., 1997 selectSet() 4 9 ≥ 111408 Hutter et al., 2002 Thornton et al., 2004 tiebreaking() 4 1 4 Hoos, 2002 Li and Huang, 2005 Prestwich, 2005 update() 5 12 ≥ 73728 Hoos, 2002 Li et al., 2007b Hutter et al., 2002 Thornton et al., 2004 Pham et al., 2008 Li and Huang, 2005 Table 3.1: SATenstein-LS components. . (a) Base-level parameters select building blocks. (b) Base-level parameters specify update mechanisms. Figure 3.1: Dependencies between SATenstein-LS parameters. Chapter 3. SATenstein-LS 19 Parameter Active When Domain Description performSearchDiversification Base level parameter {0,1} If true, block B1 is performed usePromisingList Base level parameter {0,1} If true, block B2 is performed singleClauseAsNeighbor Base level parameter {0,1} If true, block B3 is performed else, block B4 is performed selectPromVariable usePromisingList = 1 {1,...,11} See Table 3.5 heuristic singleClauseAsNeighbor = 1 {1,...,13} See Table 3.6 performAlternateNovelty singleClauseAsNeighbor = 1 {0,1 If true, performs Novelty variant with “flat move”. useAdaptiveMechanism Base level parameter {0,1} If true, uses adaptive mechanisms. adaptivenoisescheme useAdaptiveMechanism = 1 {1,2} Specifies adaptive noise mechanisms. usePromisingList = 1 adaptWalkProb useAdaptiveMechanism = 1 {0,1} If true, walk probability or diversification probability of a heuristic is adaptively tuned. performTabuSearch Base level parameter {0,1} If true, tabu variables are not considered for flipping. useClausePenalty Base level parameter {0,1} If true, clause penalties are computed. selectClause singleClauseAsNeighbor = 1 {1,2} 1 selects an UNSAT clause uniformly at random. 2 selects an UNSAT clause with a probability proportional to its clause penalty. searchDiversificationStrategy performSearchDiversification = 1 {1,2,3,4} 1 randomly selects a variable from an UNSAT clause. 2 selects the least-recently-flipped -variable from an UNSAT clause. 3 selects the least-frequently-flipped variable from an UNSAT clause. 4 selects the variable with least VW2 weight from an UNSAT clause. adaptiveProm usePromisingList = 1 {0,1} If true, performs adaptive versions of Novelty variants to select variable from promising list. adaptpromwalkprob usePromisingList = 1 {0,1} If true, walk probability or diversification adaptiveProm = 1 probability of Novelty variants used on promising list is adaptively tuned. scoringMeasure usePromisingList = 0 {1,2,3} Specifies the scoring measure. singleClauseAsNeighbor = 0 1 uses MakeCount - BreakCount 2 uses MakeCount 3 uses -BreakCount tieBreaking usePromisingList = 1 {1,2,3,4} 1 breaks ties randomly. selectPromVariable ∈ {1, 4, 5} 2 breaks ties in favor of the least-recently-flipped variable. 3 breaks tie in favor of the least-frequently-flipped variable. 4 breaks tie in favor of the variable with least VW2 score. updateSchemePromList usePromisingList = 1 {1,2,3} 1 and 2 follow G2WSAT . 3 follows gNovelty+. smoothingScheme useClausePenalty = 1 {1,2} When singleClauseAsNeighbor = 1 : 1 performs smoothing for only random 3-SAT instances with 0.4 fixed smoothing probability. 2 performs smoothing for all instances. When singleClauseAsNeighbor = 0 : 1 performs SAPS-like smoothing. 2 performs PAWS-like smoothing. Table 3.2: Categorical parameters of SATenstein-LS. Unless otherwise mentioned, for more than one parameters, the conditional operator used in the the ‘Active When’ column is AND. Chapter 3. SATenstein-LS 20 Parameter Active When Description tabuLength performTabuSearch = 1 Specifies tabu step-length. phi useAdaptiveMechanism = 1 Parameter for adaptively setting noise. singleClauseAsNeighbor = 1 theta useAdaptiveMechanism = 1 Parameter for adaptively setting noise. singleClauseAsNeighbor = 1 promPhi usePromisingList = 1 Parameter for adaptively setting noise. adaptiveProm = 1 selectPromVariable ∈ {7, 8, 9, 10, 11} promTheta usePromisingList = 1 Parameter for adaptively setting noise. adaptiveProm = {1} selectPromVariable ∈ {7, 8, 9, 10, 11} maxinc singleClauseAsNeighbor = 0 PAWS (Thornton et al., 2004) parameter for useClausePenalty = 1 additive clause weighting. smoothingScheme = 2 Table 3.3: Integer parameters of SATenstein-LS. Unless otherwise mentioned, for more than one parameters, the conditional operator used in the the ‘Active When’ column is AND. Parameter Active When Description wp singleClauseAsNeighbor = 1 Randomwalk probability for Novelty+ heuristic ∈ {2, 6, 11} useAdaptiveMechanism = 0 dp singleClauseAsNeighbor = 1 Diversification probability for Novelty++ heuristic ∈ {3, 4, 12, 13} and Novelty++′ useAdaptiveMechanism = 0 promDp usePromisingList = 1 Diversification probability for Novelty selectPromVariable ∈ {8, 10} variants used to select variable from adaptiveProm = 0 promising list novNoise singleClauseAsNeighbor = 1 Noise parameter for all Novelty variants heuristic ∈ {1, 2, 3, 4, 5, 6, 10, 11, 12, 13} useAdaptiveMechanism = 0 wpWalk singleClauseAsNeighbor = 1 Noise parameter for WalkSAT and VW1 heuristic ∈ {7, 9} useAdaptiveMechanism = 0 promWp usePromisingList = 1 Randomwalk probability for Novelty selectPromVariable ∈ {9, 11} variants used to select variable from promising list promNovNoise usePromisingList = 1 Noise parameter for all Novelty selectPromVariable ∈ 7, 8, 9, 10, 11 variants used to select variable from promising list alpha singleClauseAsNeighbor = 0 Parameter for SAPS useClausePenalty = 1 smoothingScheme = 1 rho singleClauseAsNeighbor = 0 Parameter for SAPS useClausePenalty = 1 smoothingScheme = 1 ps useClausePenalty = 1 Smoothing parameter for SAPS, RSAPS, and gNovelty+ s singleClauseAsNeighbor = 1 VW parameter for smoothing heuristic = 8 useAdaptiveMechanism = 0 c singleClauseAsNeighbor = 1 VW parameter for smoothing heuristic = 8 useAdaptiveMechanism = 0 rdp performSearchDiversification = 1 Parameter for search diversification searchDiversificationStrategy ∈ {2, 3} rfp performSearchDiversification = 1 Parameter for search diversification searchDiversificationStrategy = 4 rwp performSearchDiversification = 1 Parameter for search diversification searchDiversificationStrategy = 1 pflat singleClauseAsNeighbor = 0 Parameter for PAWS that controls useClausePenalty = 1 “flat-moves” smoothingScheme = 2 Table 3.4: Continuous parameters of SATenstein-LS. Unless otherwise mentioned, for more than one parameters, the conditional operator used in the the ‘Active When’ column is AND. Chapter 3. SATenstein-LS 21 Param. Value Design choice Based on 1 If freebie exists, use tieBreaking(); (Selman et al., 1994) else, select uniformly at random 2 Variable with best score (Li and Huang, 2005) 3 Least-recently-flipped variable (Li et al., 2007b) 4 Variable with best VW1 score (Prestwich, 2005) 5 Variable with best VW2 score (Prestwich, 2005) 6 Variable selected uniformly at random (Hoos, 1999) 7 Variable selection from Novelty (McAllester et al., 1997) 8 Variable selection from Novelty++ (Li and Huang, 2005) 9 Variable selection from Novelty+ (Hoos, 1999) 10 Variable selection from Novelty++′ (Li et al., 2007a) 11 Variable selection from Novelty+p (Li et al., 2007a) Table 3.5: Design choices for selectFromPromisingList() and corresponding values of selectPromVariable. there are two strategies for selecting a variable from promising list : choosing the variable with the highest score (Li and Huang, 2005) or choosing the least recently flipped variable (Li et al., 2007b). We added nine novel strategies based on variable selection heuristics from other solvers. These, to the best of our knowledge, have never been used before in the context of promising variable selection for G2WSAT-based algorithms. For example, in previous work, variable selection mechanisms used in Novelty variants are only applied to variables of unsatisfiable clauses, not to promising lists. Table 3.5 lists the eleven possible strategies for SelectFromPromisingList. Param. Value Selected Heuristic Dependent Parameters 1 Novelty (McAllester et al., 1997) novnoise 2 Novelty+ (Hoos, 2002) novnoise, wp 3 Novelty++ (Li and Huang, 2005) novnoise, dp 4 Novelty++′ (Li et al., 2007a) novnoise, dp 5 R-Novelty (McAllester et al., 1997) novnoise 6 R-Novelty+ (Hoos, 2002) novnoise, wp 7 VW1 (Prestwich, 2005) wpwalk 8 VW2 (Prestwich, 2005) s, c, wp 9 WalkSAT-SKC (Selman et al., 1994) wpwalk 10 Noveltyp (Li et al., 2007a) novnoise 11 Novelty+p (Li et al., 2007a) novnoise, wp 12 Novelty++p (Li et al., 2007a) novnoise, dp 13 Novelty++′p (Li et al., 2007a) novnoise, dp Table 3.6: List of heuristics chosen by the parameter heuristic and dependent pa- rameters. If promising list is empty, B2 behaves exactly like B3, which instantiates WalkSAT-based algorithms. As already described in B1, component Select- Clause() is used to select an unsatisfiable clause c. The SelectHeuristic() compo- nent selects a variable from c for flipping. Depending on a categorical parameter, SelectHeuristic() can behave like any of the thirteen well-known WalkSAT-based heuristics that include Novelty variants, VW1 and VW2. Table 3.6 lists these heuristics and related continuous parameters. We also extend the Novelty vari- Chapter 3. SATenstein-LS 22 ants with an optional “flat move” mechanism as found in the selection strategy in gNovelty+ (Thornton et al., 2004; Pham et al., 2008). Block B4 instantiates dynamic local search algorithms. The selectSet() com- ponent considers the set of variables that occur in any unsatisfied clause. It as- sociates with each such variable v a score, which depends on the clause weights of each clause that changes satisfiability status when v is flipped. These clause weights reflect the perceived importance of satisfying each clause. For example, weights might increase the longer a clause has been unsatisfied, and decrease afterwards (Hutter et al., 2002; Thornton et al., 2004). After scoring the vari- ables, selectSet() returns all variables with maximal score. Our implementation of this component incorporates three different scoring functions, including those due to McAllester et al., 1997, Selman et al., 1994, and a novel, greedier variant that only considers the number of previously unsatisfied clauses that are satis- fied by a variable flip. The tieBreaking() component selects a variable from the maximum-scoring set according to the same strategies used by the diversifica- tionStrategy() component. BlockB5 updates underlying data structures after a variable has been flipped. Performing these updates in an efficient manner is a core issue in optimizing the performance of SLS algorithms. Because the SATenstein-LS framework supports the combination of mechanisms from many different SLS algorithms, each depending on different data structures, the implementation of the update() function was technically quite challenging. 3.2 Implementation and Validation As already mentioned, SATenstein-LS is built on top of UBCSAT (Tompkins and Hoos, 2004), a well-known SLS solver development framework. UBCSAT has a novel trigger-based architecture that facilitates the reuse of existing mech- anisms. While designing and implementing SATenstein-LS, we not only studied existing SLS algorithms, as presented in the literature, but we also looked at the SAT Competition submissions of such algorithms. We found that the pseu- docode of VW2 according to Preswitch (2005) is different from its SAT Competi- tion 2005 submission, which includes a reactive mechanism. We included both versions in SATenstein-LS’s implementation. We also found that in the SAT Competition implementation of gNovelty+, Novelty uses a PAWS-like (Thorn- ton et al., 2004) “flat move” mechanism. We implemented this alternate version of Novelty in SATenstein-LS and exposed a categorical parameter to choose between the two implementations. While looking at the various source codes, we found that certain data structures are implemented in different ways. For exam- ple, different G2WSAT variants implement the update scheme of promising list in different ways. We include all such different update schemes in SATenstein-LS and declared parameter updateSchemePromList. In order to support a vast number of possible heuristics and combinations of data structures, the implementation of SATenstein-LS is fairly complicated and required the validation of individual components. For that, we compared the Chapter 3. SATenstein-LS 23 102 103 104 105 102 103 104 105 UBCSAT run−length [steps] SA Te ns te in −L S ru n− le ng th  [s tep s] (a) G2WSAT (p-value = 1) 102 103 104 105 102 103 104 105 UBCSAT run−length [steps] SA Te ns te in −L S ru n− le ng th  [s tep s] (b) adaptG2WSATp (p-value = 0.8772) 102 103 104 105 102 103 104 105 UBCSAT run−length [steps] SA Te ns te in −L S ru n− le ng th  [s tep s] (c) adaptG2WSAT+ (p-value = 1) 102 103 104 105 102 103 104 105 UBCSAT run−length [steps] SA Te ns te in −L S ru n− le ng th  [s tep s] (d) adaptNovelty+ (p-value = 1) 102 103 104 105 102 103 104 105 UBCSAT run−length [steps] SA Te ns te in −L S ru n− le ng th  [s tep s] (e) SAPS (p-value = 1) 102 103 104 105 102 103 104 105 UBCSAT run−length [steps] SA Te ns te in −L S ru n− le ng th  [s tep s] (f) RSAPS (p-value = 1) 102 103 104 105 102 103 104 105 UBCSAT run−length [steps] SA Te ns te in −L S ru n− le ng th  [s tep s] (g) PAWS (p-value = 1) 102 103 104 105 106 102 103 104 105 106 UBCSAT run−length [steps] SA Te ns te in −L S ru n− le ng th  [s tep s] (h) WalkSAT-SKC (p-value = 1) Figure 3.2: Quantile-quantile plots of the run-length distributions for SATenstein-LS vs. UBCSAT (Tompkins and Hoos, 2004) implementations of various well-known SLS solvers. Each implementation was run 5000 times on instance bw large.a with a per- run cutoff time of 600 CPU seconds. The p-value of the performed KS-test is given in the caption of each subfigure. Chapter 3. SATenstein-LS 24 performance of ten well-known algorithms implemented in UBCSAT with their SATenstein-LS equivalents. We carefully chose these ten algorithms to cover all three major classes of SLS algorithm and all the prominent mechanisms used in SLS solvers. Since SATenstein-LS and UBCSAT use the same random number generation routine, for a given instance-seed pair, UBCSAT and SATenstein-LS mostly show identical flip performance. However, the SATenstein-LS equiva- lents of some algorithms (e.g., adaptG2WSATp) do not follow algorithm execu- tion path identical to their respective UBCSAT implementations; the flip per- formance of these algorithms is not exactly identical. Figure 3.2 shows the quantile-quantile plots of the run-length distributions of UBCSAT implemen- tation and SATenstein-LS instantiation of several well-known algorithms. It demonstrates that SATenstein-LS correctly reproduces the behavior of these known algorithms, and that the underlying mechanisms have been correctly implemented in SATenstein-LS. As mentioned earlier, we implemented the 2005 SAT Competition submission of VW in SATenstein-LS, which is different from Prestwich, 2005 in its use of reactive mechanisms. Also, the 2007 SAT Competition winner gNovelty+ has not been implemented in UBCSAT, version 1.1. For these two algorithms, we compared the performance with their original SAT Competition submissions (shown in Figure 3.3. 102 103 104 105 106 102 103 104 105 106 VW run−length [steps] SA Te ns te in −L S ru n− le ng th  [s tep s] (a) VW SAT Competition 2005 submission (p- value = 0.2392) 102 103 104 105 102 103 104 105 gNovelty+ run−length [steps] SA Te ns te in −L S ru n− le ng th  [s tep s] (b) gNovelty+ SAT Competitions 2007 sub- mission (p-value = 0.6609) Figure 3.3: Quantile-quantile plots of the run-length distributions for SATenstein-LS vs. SAT Competition, 2005 submission of VW (Prestwich, 2005) and SAT Competition, 2007 submission of gNovelty+ (Pham et al., 2008). Each implementation is run 5000 times on benchmark instance bw large.a with a per-run cutoff of 600 CPU seconds. The p-value of the performed KS-test is given in the caption of each subfigure. 25 Chapter 4 Experimental Setup In order to study the effectiveness of our proposed approach for algorithm design, we configured SATenstein-LS on various problem distributions and compared the performance of the obtained SATenstein-LS solvers against that of several existing high-performance solvers on the test set for each distribution. This chapters presents the problem distributions, tuning configurations and machine specifications used in our experiments, describes the solvers we selected for comparison, and explains their selection. 4.1 Instance Distributions We considered six sets of well-known benchmark instances for SAT (see Ta- ble 4.1). These six distributions can be roughly categorized into three broad categories of SAT instances namely, industrial (CBMC(SE), FAC), handmade (QCP, SW-GCP), and random (R3SAT, HGEN). Because SLS algorithms are unable to prove unsatisfiability, we constructed our benchmark sets to include only satisfiable instances. The generators for HGEN (Hirsch, 2002) and FAC (Uchida and Watanabe, 1999) distributions generate only satisfiable instances. For the remaining distributions, we filtered out unsatisfiable instances using complete solvers as described below. For HGEN, we generated 5000 instances with the number of variables ran- domly selected from 250 to 400 (Hirsch, 2002). From these 5000 instances, we randomly chose 1000 to form a training set, and another 1000 to form a test set of the same size. For the FAC distribution, we randomly selected 2000 instances from 7140 instances generated (Uchida and Watanabe, 1999) and divided these randomly into a training and test set containing 1000 instances each. The range of the prime numbers we used was from 3000 to 4000. For QCP, we generated 30,620 instances around the solubility phase tran- sition with a satisfiable/unsatisfiable ratio of 58.7/41.3. For our QCP instance generator, we used the same parameter values given by Gomes and Selman, 1997.1 We confirmed the satisfiability status of each of the generated instances using several complete solvers. From the satisfiable instances, we randomly sampled 2000 instances and divided them randomly into a training set and test set, each containing 1000 instances. For SW-GCP, we generated 20,000 instances (Gent et al., 1999) with a satisfiable/unsatisfiable ratio of 55.9/44.1 1h×order1.55 holes, where h was randomly selected from [1.2,2.2] and order was randomly selected from [10,40]. Chapter 4. Experimental Setup 26 Distribution Description QCP SAT-encoded quasi-group completion problems (Gomes and Selman, 1997) SW-GCP SAT-encoded small-world graph-colouring problems (Gent et al., 1999) R3SAT uniform-random 3-SAT instances (Simon, 2002) HGEN random instances generated by generator HGEN2 (Hirsch, 2002) FAC SAT-encoded factoring problems (Uchida and Watanabe, 1999) CBMC(SE) SAT-encoded bounded model checking problems (Clarke et al., 2004), preprocessed by SatELite (Eén and Biere, 2005) Table 4.1: Our six benchmark distributions. and confirmed the satisfiability status of each of the generated instances using several complete solvers. We used the same parameter values given by Gent et al., 1999. The ring lattice size was randomly selected from [100,400] with ten connected nearest neighbors. We set the chromatic number to six and used a rewiring probability of 2−7. From the 20,000 SW-GCP instances thus gen- erated, we randomly sampled 2000 satisfiable instances and divided randomly into training and test sets of 1000 instances each. For R3SAT, we generated a set of 1000 instances with 600 variables and a clauses-to-variables ratio of 4.26 (instances generated with 4.26 clauses-to-variable ratio are around the solubil- ity phase transition and are considered to be hard to solve). We identified 521 satisfiable instances by running March pl (Heule and Maaren, 2007a), each of our 11 high-performance algorithms (see Table 4.4) and Kcnfs-04 (Dubois and Dequen, 2001) with cutoffs of 3600, 600 and 36000 seconds respectively.2 From these 521 instances, we randomly chose 250 to form a training set and another 250 to form a test set of the same size. Using the CBMC generator (Clarke et al., 2004), we created 611 SAT-encoded software verification instances based on a binary search algorithm with different array sizes and loop unwinding num- bers. We preprocessed these instances using SatELite (Eén and Biere, 2005). Of the instances thus obtained, we filtered out 7 unsatisfiable instances using the complete solver Minisat2.0 (Sörensson and Eén, 2007) and confirmed all the others as satisfiable. The 604 satisfiable instances thus obtained were randomly divided into a training set of 303 instances and a test set of 301 instances. 4.2 Tuning Scenario and PAR In order to perform automatic algorithm configuration using the SATenstein-LS framework, we first had to quantify performance using an objective function. 2Our complete solvers were able to rule out only 56 of the remaining 479 instances as unsatisfiable. Chapter 4. Experimental Setup 27 We chose to focus on mean runtime. However, efficient algorithm configurators cut off some runs before they terminate, making the mean ill-defined. Thus, following Hutter et al., (2007a), we define the penalized average runtime (PAR) of a set of N runs with a k-second cutoff as the mean runtime over all N runs, with capped runs counted as 10 · k seconds. Parameter Values considered tabuLength 1, 3, 5, 7, 10, 15, 20 phi 3, 4, 5, 6, 7, 8, 9, 10 theta 3, 4, 5, 6, 7, 8, 9, 10 promPhi 3, 4, 5, 6, 7, 8, 9, 10 promTheta 3, 4, 5, 6, 7, 8, 9, 10 maxinc 5, 10, 15, 20 Table 4.2: Integer parameters of SATenstein-LS and the values considered during ParamILS tuning. For each parameter, the known defaults are highlighted in bold. For parameters that are first introduced in SATenstein-LS, the default values are indicated with an underline. For our algorithm configurator, we chose the FocusedILS procedure from the ParamILS framework, version 2.2 (Hutter et al., 2008, 2007b), because it is the only approach of which we are aware that is able to operate effectively on extremely large, discrete parameter spaces. We set k to five seconds, and allotted 2 days to each run of FocusedILS. FocusedILS takes as input a param- eterized algorithm (target algorithm), a specification of the domains for all the parameters, a set of training instances and an evaluation metric. FocusedILS outputs a parameter configuration of the target algorithm that approximately minimizes the evaluation metric. Since ParamILS cannot operate on contin- uous parameters, each continuous parameter is discretized between 3 and 16 values that seemed reasonable to us (see Table 4.3), always including the known defaults. Except for few parameters (e.g., s,c) for which we used the same dis- crete domain as mentioned in the paper from the original author (Prestwich, 2005), we mostly selected values using a regular grid and a range that seemed reasonable to us. For integer parameters, we specified 4 to 10 values for each of them, always including the known defaults (see Table 4.2). Categorical pa- rameters and their respective domains are listed in Table 3.2. The total size of the space is 4.82 × 1012. Since the performance of FocusedILS can vary significantly depending on the order in which instances appear in the training set, we ran FocusedILS 10 times on the training set, using different, randomly determined instance orderings for each run. From the 10 parameter configura- tions obtained from FocusedILS for each instance distribution D, we selected the parameter configuration with the best penalized average runtime on the training set. We then evaluated this configuration on the test set. For a given distribution D, we refer to the corresponding instantiation of SATenstein-LS as SATenstein-LS[D]. Chapter 4. Experimental Setup 28 Parameter Discrete Values Considered wp 0, 0.01, 0.03, 0.04, 0.05, 0.06, 0.07, 0.1, 0.15, 0.20 dp 0.01, 0.03, 0.05, 0.07, 0.1, 0.15, 0.20 promDp 0.01, 0.03, 0.05, 0.07, 0.1, 0.15, 0.20 novNoise 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 , 0.8 wpWalk 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 , 0.8 promWp 0.01, 0.03, 0.05, 0.07, 0.1, 0.15, 0.20 promNovNoise 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 , 0.8 alpha 1.01, 1.066, 1.126, 1.189, 1.3, 1.256, 1.326, 1.4 rho 0, 0.17, 0.333, 0.5, 0.666, 0.8, 0.83, 1 s 0.1, 0.01, 0.001 c 0.1, 0.01, 0.001, 0.0001, 0.00001, 0.000001 rdp 0.01, 0.03, 0.05, 0.07, 0.1, 0.15 rfp 0.01, 0.03, 0.05, 0.07, 0.1, 0.15 rwp 0.01, 0.03, 0.05, 0.07, 0.1, 0.15 pflat 0.05, 0.10, 0.15, 0.20 ps 0, 0.033, 0.05, 0.066, 0.1, 0.133, 0.166, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0 Table 4.3: Values considered for continuous parameters of SATenstein-LS during ParamILS tuning. For each parameter, the known defaults are highlighted in bold. For parameters that are first introduced in SATenstein-LS, the default values are indicated with an underline. 4.3 Solvers Used for Performance Comparison For each distribution D, we compared the performance of SATenstein-LS[D] against that of 11 high-performance SLS-based SAT solvers on the test set. We included every SLS algorithm that won a medal in any category of the SAT Competition in the last five years (SAT Competition, 2009).3Although dynamic local search (DLS) algorithms did not win any SAT Competition medals, we also included three prominent, high-performing DLS algorithms for the follow- ing reasons. First, some of them represented the state of the art when introduced (e.g., SAPS (Hutter et al., 2002)) and still offer competitive performance on many instances. Secondly, techniques used in these algorithms are incorporated in other recent high-performance SLS algorithms. For example, the additive clause weighting scheme used in PAWS is also used in the 2007 SAT Competi- tion winner gNovelty+ (Pham et al., 2008). Finally, all the medal-winning SLS algorithms won medals in the random satisfiable category. In our experiments, we have also used problem distributions from the other two categories (hand- made, industrial) where these solvers may not represent the state-of-the-art SLS 3Although we cite 2009 SAT Competition winners at some places, our experiments were conducted before the 2009 SAT Competitions took place. Hence, we compare solvers up to 2007. Chapter 4. Experimental Setup 29 solvers. Overall, we selected what we believe to be the most representative high- performance solvers from each SLS solver category. We call these algorithms challengers and list them in Table 4.4. SATenstein-LS can be instantiated such that it emulates all 11 of these challenger algorithms, except for preprocessing components used by three of the challengers (Ranov, G2 and AG20). However, in some cases the original implementations of these algorithms are more efficient (on our data, by at most about a factor of two). This is because SATenstein-LS’s generality rules out some data structure optimizations. Thus, we based all of our experimental comparisons on the original algorithm implementations, as submitted to the respective SAT Competitions. The exceptions are PAWS, to which the UBCSAT implementation is almost identical to the original in terms of runtime, SAPS, RSAPS, and ANOV, for which the UBCSAT implementations are those used in the competitions. All of our comparisons on the test set are based on running each solver 10 times, with a per-run cutoff of 600 seconds. Algorithm Short Name Reason for Inclusion Ranov Ranov gold 2005 SAT Competition (random) G2WSAT G2 silver 2005 SAT Competition (random) VW VW bronze 2005 SAT Competition (random) gNovelty+ GNOV gold 2007 SAT Competition (random) adaptG2WSAT0 AG20 silver 2007 SAT Competition (random) adaptG2WSAT+ AG2+ bronze 2007 SAT Competition (random) adaptNovelty+ ANOV gold 2004 SAT Competition (random) adaptG2WSATp AG2p performance comparable to adaptG2WSAT+, G2WSAT, and Ranov; see (Li et al., 2007a) SAPS SAPS prominent DLS algorithm RSAPS RSAPS prominent DLS algorithm PAWS PAWS prominent DLS algorithm Table 4.4: Our eleven challenger algorithms. Category Solver Reason for Inclusion Industrial Picosat gold, silver (CBMC(SE) and FAC) (Biere, 2008, 2007) 2007 SAT Competition (industrial) Minisat2.0 bronze, silver (Sörensson and Eén, 2007) 2007 SAT Competition (industrial) Handmade Minisat2.0 bronze, silver (QCP and SW-GCP) (Sörensson and Eén, 2007) 2007 SAT Competition (handmade) March pl Improved, bug-free version of (Heule and Maaren, 2007a) March ks (Heule and Maaren, 2007b), winner of gold in 2007 SAT Competition (handmade) Random Kcnfs 04 silver (HGEN and R3SAT) (Dubois and Dequen, 2001) 2007 SAT Competition (random) March pl Improved, bug-free version of (Heule and Maaren, 2007a) March ks (Heule and Maaren, 2007b), winner of silver in 2007 SAT Competition (random) Table 4.5: Complete solvers we compared against. Our goal was to improve the state of the art in SAT solving. Thus, although SATenstein-LS can be only instantiated as SLS solvers, we also compared its performance to that of complete solvers (listed in Table 4.5) that are state of Chapter 4. Experimental Setup 30 the art for industrial distributions. Unlike SLS solvers, the complete solvers are deterministic. Thus, on every instance in each distribution, we ran each complete solver once with a per-run cutoff of 600 seconds. 4.4 Execution Environment We carried out our experiments on a cluster of 55 dual 3.2GHz Intel Xeon PCs with 2MB cache and 2GB RAM, running OpenSuSE Linux 10.1. Our computer cluster was managed by a distributed resource manager, Sun Grid Engine soft- ware (version 6.0). Runtimes for all algorithms (including FocusedILS) were measured as CPU time on these reference machines. Each run of any solver only used one CPU. 31 Chapter 5 Results In this chapter, we present the results of performance comparisons of SATenstein-LS against our 11 challenger SLS solvers (listed in Table 4.4), two state-of-the-art complete solvers (see Table 4.5), and a hypothetical Oracle that always picks an SLS solver with minimum medium runtime on a per-instance basis. Although in our tuning experiments we optimized SATenstein-LS for Penalized Average Runtime (PAR), we also examine its performance in terms of other performance metrics such as median-of-median runtime and percentage of instances solved within cutoff time. We describe each SATenstein-LS solver found in terms of the challengers that it most closely resembles. We also present a distance metric between two SATenstein-LS configurations and use it to compute the distance between each SATenstein-LS solver found and the best SATenstein-LS equiv- alent within the parameter space of each challenger. 5.1 Comparison with Challengers In every one of our six benchmark distributions, we were able to find a SATenstein-LS configuration that outperformed all 11 challengers. Our results are summarized in Table 5.1. In terms of penalized average runtime, the performance metric we explic- itly optimized using ParamILS (evaluated using a different cutoff of 600 sec- onds during testing), our SATenstein-LS solvers achieved better performance than every challenger on every distribution. For QCP, HGEN, and CBMC(SE), SATenstein-LS achieved a PAR that was orders of magnitude better than the respective best challengers. For SW-GCP, R3SAT, and FAC there was sub- stantial, but less dramatic improvement. The modest improvement in R3SAT was not very surprising; R3SAT is a well-known SAT distribution on which SLS solvers for the past 17 years have been evaluated and optimized. Con- versely, on a new benchmark distribution, CBMC(SE), where DPLL solvers are considered to represent the state of the art, SATenstein-LS solvers performed markedly better than every challenger and narrowed the huge performance gap between DPLL solvers and SLS solvers on this distribution. We were surprised to see the amount of improvement we obtained for HGEN, a hard random SAT distribution very similar to R3SAT, and QCP, a widely-known SAT distribu- tion. We noticed that in HGEN, some older solvers such as SAPS and PAWS performed much better than recent SAT Competition medal winners such as GNOV and AG20. Also, for QCP, a relatively older algorithm, ANOV was the best Chapter 5. Results 32 Solvers QCP SW-GCP R3SAT HGEN FAC CBMC(SE) 0.13 0.03 1.51 0.03 12.22 5.59 SATenstein-LS[D] 0.01 0.03 0.14 0.02 8.03 0.02 (KhudaBukhsh et al., 2009) 100% 100% 100% 100% 100% 100% 422.33 0.24 10.93 52.87 5912.99 2238.7 GNOV 0.03 0.09 0.14 0.73 N/A 0.75 (Pham et al., 2008) 92.7% 100% 100% 99.4% 0% 61.13% 1051 0.62 2.37 139.33 3607.4 2170.67 AG20 0.03 0.12 0.14 0.57 N/A 0.67 (Li et al., 2007a) 80.5% 100% 100% 98% 30.2% 61.13% 1080.29 0.45 3.3 138.84 1456.4 2161.59 AG20 0.03 0.08 0.16 0.61 237.50 0.77 (Li et al., 2007b) 80.3% 100% 100% 97.8% 84.2% 61.13% 76.22 0.15 14.14 156.96 942.26 1231.01 RANOV 0.1 0.12 0.32 0.95 155.77 1231.01 (Pham and Anbulagan, 2007) 98.7% 100% 100% 97.7% 92.1% 79.73% 2952.56 4103.27 5.32 110.02 5944.6 2150.31 G2 361.21 N/A 0.13 0.61 N/A 0.68 (Li and Huang, 2005) 80.3% 100% 100% 97.8% 84.2% 61.13% 1025.9 159.67 9.53 177.9 3411.93 385.73 VW 2952.56 40.96 0.75 3.23 N/A 0.27 (Prestwich, 2005) 82.2% 98.9% 100% 97.5% 31.7% 92.69% 28.3 0.06 12.4 147.53 3258.66 2081.94 ANOV 0.01 0.03 0.21 0.76 N/A 5.81 (Hoos, 2002) 99.6% 100% 100% 97.6% 37.2% 61.79% 1104.42 0.45 2.38 107.4 1989.91 2282.37 AG2p 0.02 0.07 0.13 0.49 315.48 3.18 (Li et al., 2007b) 79.4% 100% 100% 98.4% 72.5% 61.13% 1256.2 3872.08 22.81 48.31 17.39 613.15 SAPS 0.03 N/A 1.80 3.00 11.60 0.04 (Hutter et al., 2002) 79.2% 33.2% 100% 99.5% 100% 90.03% 1265.37 5646.39 14.81 38.51 19.39 794.93 RSAPS 0.04 N/A 2.13 2.44 12.88 0.03 (Hutter et al., 2002) 78.4% 5% 100% 99.7% 100% 85.38% 1144.2 4568.59 2.4 73.27 26.51 1717.79 PAWS 0.02 N/A 0.12 0.96 12.62 19.99 (Thornton et al., 2004) 80.8% 22.1% 100% 99.2% 99.9% 68.77% Table 5.1: Performance summary of SATenstein-LS and the 11 challengers. Every algorithm was run 10 times on each instance with a cutoff of 600 CPU seconds per run. Each cell 〈i, j〉 summarizes the test-set performance of algorithm i on distribution j as a/b/c, where a (top) is the the penalized average runtime; b (middle) is the median of the median runtimes over all instances (not defined if fewer than half of the median runs failed to find a solution within the cutoff time); c (bottom) is the percentage of instances solved (i.e., having median runtime < cutoff). The best-scoring algorithm(s) in each column are indicated in bold, and additionally the best-scoring challenger(s) are indicated with an underline. Chapter 5. Results 33 challenger. These observations led us to believe that the strong performance of SATenstein-LS was partly due to the fact that the past seven years of SLS SAT solver development did not take these types of distributions into account, nor yielded across-the-board improvements. We also evaluated the performance of SATenstein-LS solvers using other performance metrics like median-of-median runtime and percentage of solved instances. If a solver finishes most of the runs on most instances, the capped runs will not affect its median-of-median performance. Thus the advantage of this metric is that it removes the dependence on the penalty for timed-out runs in used in the calculation of PAR. Table 5.1 shows that although our SATenstein-LS solvers are obtained by optimizing for PAR, SATenstein-LS solvers outperformed every challenger in every distribution except for R3SAT, in which the challengers achieved slightly better performance than SATenstein-LS. Finally, we measured the percentage of instances on which the median run- time was below the cutoff used for capping runs. According to this measure, SATenstein-LS either equaled or beat the best challenger, since it solved 100% of the instances in every benchmark distribution. In contrast, only 3 challengers managed to solve more than 40% of the instances in every distribution. Over- all, SATenstein-LS solvers did well on these measures that were not explicitly optimized. The relative performance of the challengers varied significantly across differ- ent distributions. For example, the three dynamic local search solvers (SAPS, PAWS, and RSAPS) performed substantially better than the other challengers on factoring instances (FAC). However, on SW-GCP, their relative performance is weak. Similarly, GNOV (SAT Competition 2007 winner in the random sat- isfiable category) performed very poorly in our two industrial benchmark dis- tributions, CBMC(SE) and FAC, but solved SW-GCP and HGEN instances quite efficiently.1 This suggests that different distributions are most efficiently solved by rather different solvers. Our automatic algorithm construction pro- cess found solvers that outperformed every challenger on each distribution. This result clearly demonstrates that the vast design space of candidate solvers built from combining components gleaned from existing high-performance algorithms contains better solvers than those previously known, and that the automatic exploration of this huge space can effectively find such improved designs. Of course, solvers optimized for individual benchmark distributions could in prin- ciple be combined using an instance-based algorithm selection technique like SATzilla (Nudelman et al., 2004a; Xu et al., 2008), yielding even stronger per- formance. So far, we have discussed performance metrics that describe aggregate per- formance over the entire test set. One might wonder whether a possible reason for the strong performance of SATenstein-LS was that its efficiency in solving a few instances compensating for poor performance on other instances. Table 5.2 summarizes the compared performance of each SATenstein-LS solver with each 1Interestingly, on both types of random instances we considered, GNOV failed to outperform some of the older solvers, in particular, PAWS and RSAPS. Chapter 5. Results 34 10-2 10-1 100 101 102 103 10-2 10-1 100 101 102 103 SATenstein-LS[R3SAT] median runtime (CPU sec) P A W S  m ed ia n ru nt im e (C P U  s ec )   Figure 5.1: Scatter plot of median runtimes of SATenstein-LS[R3SAT] vs PAWS on the test set for R3SAT. Every algorithm was run 10 times on each instance with a cutoff of 600 CPU seconds per run. challenger on a per-instance basis, and shows that this is typically not the case. Except for R3SAT, SATenstein-LS solvers outperformed the respective best challengers for each distribution. On R3SAT, PAWS was the challenger that outperformed SATenstein-LS most frequently (62%). Figure 5.1 shows that the performance of PAWS and SATenstein-LS were highly correlated, and that, while instances that are easy for both algorithms tended to be solved faster by PAWS, SATenstein-LS performed better on harder instances. We observe the same qualitative trend for other challengers on R3SAT. Challengers QCP SW-GCP R3SAT HGEN FAC CBMC(SE) GNOV 99 100 55 100 100 100 AG20 87 98 55 100 100 100 AG20 87 97 64 96 100 100 RANOV 100 100 96 99 100 100 G2 91 100 44 100 100 100 VW 92 100 100 100 100 100 ANOV 83 63 72 99 100 100 AG2p 81 93 55 100 100 100 SAPS 82 100 99 100 73 80 RSAPS 79 100 99 100 78 75 PAWS 79 100 38 100 80 100 Table 5.2: Percentage of instances on which SATenstein-LS achieved better median runtime than each of the 11 challengers. Medians were taken over 10 runs on each instance with a cutoff time of 600 CPU seconds per run. When both SATenstein-LS and challenger solved a given instance and achieved indistinguishable median runtimes, we counted that instance as 0.5 for SATenstein-LS and 0.5 for the challenger. Our penalized average runtime measure depends on the choice of test cutoff, which sets the penalty. For example, if an algorithm solves all the instances Chapter 5. Results 35 10−1 100 101 102 103 0 0.2 0.4 0.6 0.8 1 Cutoff time (CPU seconds) PA R Training cutoff Test cutoff SATenstein−LS ANOV RANOV GNOV (a) SW-GCP 10−1 100 101 102 103 0 50 100 150 200 250 300 350 400 450 Cutoff time (CPU seconds) PA R Training cutoff Test cutoff SATenstein−LS ANOV RANOV GNOV (b) QCP 10−1 100 101 102 103 0 1 2 3 4 5 6 7 8 9 Cutoff time (CPU seconds) PA R Training cutoff Test cutoff SATenstein−LS AG20 AG2p PAWS (c) R3SAT 10−1 100 101 102 103 0 10 20 30 40 50 60 70 80 Cutoff time (CPU seconds) PA R Training cutoff Test cutoff SATenstein−LS RSAPS SAPS GNOV (d) HGEN 10−1 100 101 102 103 0 10 20 30 40 50 60 70 80 Cutoff time (CPU seconds) PA R Training cutoff Test cutoff SATenstein−LS SAPS RSAPS PAWS (e) FAC 10−1 100 101 102 103 0 100 200 300 400 500 600 700 800 900 Cutoff time (CPU seconds) PA R Training cutoff Test cutoff SATenstein−LS VW SAPS RSAPS (f) CBMC(SE) Figure 5.2: Penalized average runtime for our SATenstein-LS solver and the top three challengers for each distribution, as a function of cutoff, based on 10 runs on each test-set instance. Chapter 5. Results 36 but does not solve many of them quickly, then it will score well if the cutoff is large, but it will have weaker PAR score for lower cutoffs. We found it reassuring that this problem did not arise in the SATenstein-LS solvers we considered. In Figure 5.2, we plot the PAR score of the SATenstein-LS solvers and the top three challengers in each distribution. We see that, while the choice of cutoff increases, the PAR increases (since the penalty for not solving an instance increases) and then decreases (since for solved instances, the penalty gets replaced by the actual runtime). Once all instances on a given test set are solved, PAR remains constant. Figure 5.2 shows that regardless of the choice of cutoff, SATenstein-LS always outperformed the top three challengers in every distribution (the only exception being in R3SAT, where AG20 had a marginally better PAR score than SATenstein-LS within a very small range of cutoff times). 5.2 Comparison with a Hypothetical Oracle We have already demonstrated that, when compared against an individual challenger, our SATenstein-LS solvers achieved better performance than every challenger in every benchmark distribution. Figure 5.3 explores the potential of SATenstein-LS solvers further by comparing them against a hypothetical Oracle that picks the challenger with best runtime on a per-instance basis without any overhead. Note that such an Oracle does not exist in the real world. Figure 5.3 clearly shows that the performance of SATenstein-LS[D] was comparable with that of Oracle[D] in every distribution, though slightly worse in R3SAT. For QCP, HGEN, and CBMC(SE), SATenstein-LS[D] performed much better than the respective Oracle solvers. Figure 5.3 also shows a gen- eral qualitative trend — on most distributions, in paricular for SW-GCP, QCP, and CBMC(SE), the performance advantage of SATenstein-LS over the Ora- cle seems to increase with instance hardness. For a given distribution D, the performance of Oracle[D] is the upper bound of performance achieved by any algorithm-selection-based method using a given set of algorithms. Note that SATenstein-LS is not an algorithm selection approach. Our results suggest that our SATenstein-LS solvers found designs that can be better than a per- instance algorithm selection approach and that our new solvers are very likely to perform favorably against any algorithm portfolio constructed by using the challengers. 5.3 Comparison with Complete Solvers Table 5.3 summarizes the comparison of performance results between SATenstein-LS solvers and four complete solvers (two for each distribution). On four of our six benchmark distributions, SATenstein-LS solvers comprehensively outper- formed the complete solvers. For the two industrial distributions we considered, Chapter 5. Results 37 10−2 10−1 100 101 102 103 10−2 10−1 100 101 102 103 SATenstein−LS median runtime (CPU sec) O ra cle  m ed ia n ru nt im e (C PU  se c) SOLVED BY BOTH (a) SW-GCP 10−2 10−1 100 101 102 103 10−2 10−1 100 101 102 103 SATenstein−LS median runtime (CPU sec) O ra cle  m ed ia n ru nt im e (C PU  se c) SOLVED BY BOTH SOLVED BY ONE (b) QCP 10−2 10−1 100 101 102 103 10−2 10−1 100 101 102 103 SATenstein−LS median runtime (CPU sec) O ra cle  m ed ia n ru nt im e (C PU  se c) SOLVED BY BOTH (c) R3SAT 10−2 10−1 100 101 102 103 10−2 10−1 100 101 102 103 SATenstein−LS median runtime (CPU sec) O ra cle  m ed ia n ru nt im e (C PU  se c) SOLVED BY BOTH SOLVED BY ONE (d) HGEN 10−2 10−1 100 101 102 103 10−2 10−1 100 101 102 103 SATenstein−LS median runtime (CPU sec) O ra cle  m ed ia n ru nt im e (C PU  se c) SOLVED BY BOTH (e) FAC 10−2 10−1 100 101 102 103 10−2 10−1 100 101 102 103 SATenstein−LS median runtime (CPU sec) O ra cle  m ed ia n ru nt im e (C PU  se c) SOLVED BY BOTH SOLVED BY ONE (f) CBMC(SE) Figure 5.3: Scatter plot of median runtimes of SATenstein-LS vs Oracle on the test set for each distribution. Chapter 5. Results 38 Solvers QCP SW-GCP R3SAT HGEN FAC CBMC(SE) 0.13 0.03 1.51 0.03 12.22 5.59 SATenstein-LS[D] 0.01 0.03 0.14 0.02 8.03 0.02 100% 100% 100% 100% 100% 100% 35.05 2.17 0.03 0.23 Minisat2.0 0.02 0.9 0.02 0.03 99.5% 100% 100% 100% 0.02 0.03 Picosat 0.02 0.01 100% 100% 120.29 253.99 3543.01 2763.41 March pl 0.2 1.12 N/A 400.78 98.1% 95.8% 42% 55.2% 4905.60 3108.77 Kcnfs 04 N/A N/A 18.8% 49.5% Table 5.3: Performance summary of SATenstein-LS and the complete solvers. Every complete solver was run once (SATenstein-LS was run 10 times) on each instance with a per-run cutoff of 600 CPU seconds. Each cell 〈i, j〉 summarizes the test-set perfor- mance of algorithm i on distribution j as a/b/c, where a (top) is the the penalized av- erage runtime; b (middle) is the median runtime over all instances (for SATenstein-LS, it is the median of the median runtimes over all instances ); c (bottom) is the per- centage of instances solved. The best-scoring algorithm(s) in each row are indicated in bold, and additionally the best-scoring complete solver(s) are indicated with an underline. An empty cell, 〈i, j〉, means we did not run complete solver i on the test set of distribution j. we found the performance of the complete solvers to be much better than that of the SATenstein-LS solvers and any known local search solver. The strong results on our four benchmark distributions did not come as a surprise, since we primarily chose to compare against the complete solvers because they are known to perform much better than SLS solvers on industrial instances. The success of DPLL solvers on industrial instances is partly attributed to their abil- ity to take advantage of instance structure (e.g., clause learning). With current technology, we find that local search solvers cannot compete with state-of-the- art DPLL solvers on industrial instances. However, SATenstein-LS solvers have made significant progress in closing the gap. For example, for CBMC(SE), com- plete solvers were five orders of magnitude better than the previously known best SLS solver, VW. SATenstein-LS reduced the performance gap to three orders of magnitude. We also obtained some modest improvements (a factor of 1.42) for the FAC distribution. Chapter 5. Results 39 5.4 Heterogeneous Mixtures of Instance Distributions So far, we have seen that for homogeneous distributions, our SATenstein-LS solvers performed extremely well. To test our approach on heterogenous distri- butions, specifically, we configured SATenstein-LS on four distributions made up of instances from two or more homogeneous distributions (see Table 5.4). In order to reduce the effect of having different numbers of training instances, for each heterogenous distribution, we constructed the training set in the following way. We first select a component distribution uniformly at random. From the component distribution, we then select a problem instance uniformly at ran- dom. We continue this process till we gather 2000 training instances. For the test set, we combine all the instances from the component homogeneous distri- butions. We used the same tuning scenario as for our homogenous distributions and evaluated the performance of our newly obtained solvers in the same way (see Chapter 4). Heterogenous Distribution Component Distributions Random R3SAT and HGEN Handmade QCP and SW-GCP Industrial FAC and CBMC(SE) Big-mix R3SAT, HGEN, QCP, FAC, SW-GCP and CBMC(SE) Table 5.4: Description of heterogenous distributions we used. In each case, we sampled uniformly at random from the component homogeneous distributions. Dist SATenstein-LS[D] Best Challenger Handmade 0.11 ANOV : 14.18 Random 19.71 RSAPS : 33.77 Industrial 16.21 SAPS : 155.23 Big-mix 253 RANOV : 340.73 Table 5.5: Performance summary of SATenstein-LS and the best challenger. Every algorithm was run 10 times on each instance with a cutoff of 600 CPU seconds per run. Each cell 〈i, j〉 summarizes the test-set penalized average runtime (PAR) of algorithm i on distribution j. Table 5.5 summarizes the performance of SATenstein-LS solvers and the respective best challengers in each of our four mixed distributions. For Hand- made and Industrial, SATenstein-LS solvers performed much better than their respective best challengers: we obtained, respectively, two and one order(s) of magnitude performance improvements. For Random, we obtained a con- siderable improvement by a factor of 1.71. We also obtained a 34% overall improvement over the best challenger, RANOV. Chapter 5. Results 40 We also looked at how these new solvers performed on individual homoge- nous distributions (see Table 5.6). We found that there was a performance trade- off between HGEN and R3SAT instances; none of the SATenstein-LS[Random] and SATenstein-LS[Big-mix] could do well on both distributions. We also found that the high PAR score of SATenstein-LS[Big-mix] was mainly due to its poor performance in QCP and CBMC(SE). Dist SATenstein-LS[Category] SATenstein-LS[Big-mix] QCP 0.17 915.81 SW-GCP 0.07 0.27 R3SAT 97.38 1.55 HGEN 0.29 31.96 FAC 17.35 13.27 CBMC(SE) 12.43 640.82 Table 5.6: Performance summary of SATenstein-LS obtained for each cate- gory (Handmade, Random, and Industrial) and the Big-mix distribution on a per- distribution basis. Every algorithm was run 10 times on each instance with a cutoff of 600 CPU seconds per run. Each cell 〈i, j〉 summarizes the test-set penalized average runtime (PAR) of algorithm i on distribution j. Distribution Using Block Most Similar to QCP 1,3,5 VW, R-Novelty+, AG20 SW-GCP 1,3 G2, Novelty++ HGEN 1,3,5 G2, VW, ANOV R3SAT 4, 5 RSAPS with a different tie-breaking mechanism FAC 4,5 SAPS with a different way of scoring variables CBMC(SE) 1,4,5 GNOV and RSAPS Random 1,2,5 AG2+ and VW Handmade 2, 5 G2, that uses Novelty++′ for promising variable selection Industrial 4, 5 SAPS with a different tie-breaking mechanism Big-mix 4, 5 SAPS with a different way of scoring variables Table 5.7: SATenstein-LS configurations. 5.5 Configurations Found To better understand the SATenstein-LS[D] solvers, we compared them with the SATenstein-LS instantiations corresponding to each challenger (shown in Table 5.7). SATenstein-LS[QCP] uses blocks 1, 3, and 5 and an adaptive pa- rameter update mechanism similar to that in AG2+. Recall that block 1 is used for performing search diversification and block 5 is used to update data structures, tabu attributes and clause penalties. In block 3, which is used to in- stantiate a solver belonging to WalkSAT architecture, selectHeuristic is based on R-Novelty+, and in block 1, diversification flips the variable with minimum vari- Chapter 5. Results 41 able weight as in VW1 (Prestwich, 2005). SATenstein-LS[SW-GCP] uses blocks 1 and 3, resembling Novelty++ as used within G2. SATenstein-LS[R3SAT] uses blocks 4 and 5; it is closest to RSAPS, but uses a different tie-breaking mecha- nism. Recall that block 4 is used to instantiate dynamic local search algorithms. SATenstein-LS[HGEN] uses blocks 1, 3, and 5. In block 1 it is similar to G2 and in block 3 is closest to VW. In block 5 it uses the same adaptive parameter update mechanism as ANOV. SATenstein-LS[FAC] uses blocks 4 and 5; its instantiation closely resembles that of SAPS, but differs in the way the score of a variable is computed. SATenstein-LS[CBMC(SE)] uses blocks 1, 4, and 5, drawing on elements of GNOV and RSAPS. SATenstein-LS[Random] uses blocks 1, 2 and 5. It is closest to VW and AG2+. It selects the promising variable with least VW2 score and uses VW1 as its stochastic component. In the update block, it uses reactive mechanisms and also employs a tabu-length of 1. SATenstein-LS[Handmade] uses blocks 2 and 5. It is closest to G2, but uses a different stochastic component (R-Novelty+). It uses Novelty++′, one of our novel mechanisms for promising variable selection. SATenstein-LS[Industrial] uses blocks 4 and 5; it is clos- est to SAPS, but uses a different tie-breaking mechanism. SATenstein-LS[Big-mix] uses blocks 4 and 5; it is closest to SAPS, but differs in the way the score of a variable is computed. It also considers tabu status in its update mechanism. Interestingly, only two of the ten configurations we found uses block 2 that uses promising list, the block used by all the recent SAT Competition winners. This indicates that many interesting designs that could compete with existing high-performance solvers still remain unexplored in SLS design space. Also, we find that some of the SATenstein-LS configurations (e.g., SATenstein-LS[R3SAT], SATenstein-LS[FAC], and SATenstein-LS[Industrial]) are indeed very close to existing SLS solvers (e.g., SAPS and RSAPS) yet perform much better than the algorithms they closely resemble. This shows how even subtle variations in design may lead to substantial performance improvements. This also underscores the importance of an automated approach, since manually finding such subtle changes is very difficult. The full parameter configurations of these algorithms can be found in Table 5.9. To understand how different our solvers are from existing solvers, we de- fine a new distance metric that measures the degree of similarity between a SATenstein-LS configuration (denoted as B) and a challenger. We define our distance function in the following way. First, we find the equivalent SATenstein-LS configuration for the challenger. Let this configuration be denoted as A. Then we arrange all active parameters of A in a hierarchical fashion. Level 0 param- eters are not conditioned on any other parameters. Any level i parameter is a parameter that is conditioned on the values of one or more level i-1 parameters. Next, for each level, we compare all the active parameters in A with those in B. The distance for a given level is the percentage of unmatched active parameters of A. A 100% mismatch at a given level indicates 100% mismatch at subsequent higher levels. The distance between A and B is expressed by a vector [a0, a1, ..., aj ], where (j + 1) is the total number of levels present in the hierarchy of all active parameters of A and ak denotes the level k distance between A and B. Table 5.8 presents the distance between the SATenstein-LS solvers and the Chapter 5. Results 42 challengers. For a given distributionD, the closest challenger to SATenstein-LS[D] is the challenger that has the minimum level 0 distance. Ties are broken in favor of the challenger with the lesser level 1 distance. Similarly, any remain- ing ties are broken in favor of the challenger with the lesser level 2 distance. We find that our distance metric reasonably matches our design-level descrip- tions of the SATenstein-LS solvers. For example, for R3SAT, FAC, SW-GCP, Industrial, and Big-mix, the design-level descriptions of the new solvers com- pletely matched with our computed distances. For some distributions there were ties between two or more algorithms. For example, the distances between SATenstein-LS[Handmade] and AG20, AG2p, and AG2+ were identical. This is expected because AG20, AG2p, and AG2+ are very similar algorithms and differ by just one parameter (heuristic). For a given level, we treat all parameters equally when we compute the distance. However, our design-level description is often guided by our perceived importance of certain parameters over others. Because of this, our design-level description did not match with our distance metric for some distributions. For example, SATenstein-LS[QCP] is closest to VW according to the design-level description. However, according to our distance metric, AG20, AG2p, and AG2+ are the closest challengers. Chapter 5. Results 43 Distribution GNOV AG20 AG2+ G2 VW ANOV AG2p SAPS RSAPS PAWS 33 16 16 33 16 16 16 66 50 66 QCP 55 37 37 25 50 60 37 0 0 0 0 0 0 0 0 0 0 0 0 0 50 33 33 16 33 33 33 50 66 50 SW-GCP 11 12 12 25 50 20 12 0 0 0 0 0 0 0 0 0 0 0 0 0 50 50 50 66 33 33 50 16 0 16 R3SAT 0 50 50 0 0 20 50 33 33 66 0 0 0 0 0 0 0 75 100 0 33 16 16 33 16 16 16 66 50 66 HGEN 44 50 50 25 50 80 50 0 0 0 0 0 0 0 0 0 0 0 0 0 66 66 66 50 50 50 66 0 16 0 FAC 0 0 0 0 0 0 0 33 33 66 0 0 0 0 0 0 0 100 100 0 33 50 50 66 50 50 50 33 16 33 CBMC(SE) 0 50 50 0 0 20 50 0 0 33 0 0 0 0 0 0 0 75 100 0 66 66 66 50 50 50 66 0 16 0 Industrial 0 0 0 0 0 0 0 33 33 66 0 0 0 0 0 0 0 100 100 0 50 16 16 0 33 33 16 50 66 50 Handmade 33 25 25 50 50 20 25 0 0 0 0 0 0 0 0 0 0 0 0 0 33 16 16 33 50 50 16 100 83 100 Random 77 62 62 50 50 60 62 0 0 0 0 0 0 0 0 0 0 0 0 0 83 83 83 66 66 66 83 16 33 16 Big-mix 0 0 0 0 0 0 0 33 33 66 0 0 0 0 0 0 0 100 100 0 Table 5.8: Distance between SATenstein-LS[D] and each challenger. Each cell 〈i, j〉 summarizes the distance between challenger j and SATenstein-LS[i] as a/b/c, where a (top) is the level 0 distance; b (middle) is the level 1 distance; c (bottom) is the level 2 distance. For each distribution, the closest algorithm distances(s) are highlighted in bold. Chapter 5. Results 44 Distribution Parameter Configuration singleClauseAsNeighbor = 1, performSearchDiversification = 1, searchDiversificationStrategy = 4, useAdaptiveMechanism = 0, QCP useClausePenalty = 0, usePromisingList = 0, heuristic=5, performTabuSearch = 0, scoringMeasure = 1, rfp = 0.07, selectClause = 1 singleClauseAsNeighbor = 1, performSearchDiversification = 1, searchDiversificationStrategy = 3, useAdaptiveMechanism = 0, SW-GCP useClausePenalty = 0, usePromisingList = 0, heuristic = 1 performTabuSearch = 0, novNoise = 0.1 , rdp=0.01, selectClause = 1 singleClauseAsNeighbor = 0, performSearchDiversification = 0, useAdaptiveMechanism = 1, usePromisingList = 0, R3SAT performTabuSearch = 0, useClausePenalty = 1, selectClause = 1, tieBreaking = 2, scoringMeasure = 1, alpha = 1.126, wp = 0.04, smoothingScheme = 1, rho=0.17, singleClauseAsNeighbor = 1, performSearchDiversification = 1, searchDiversificationStrategy = 3, useAdaptiveMechanism = 1, HGEN performTabuSearch = 0, usePromisingList = 0, adaptiveProm = 0, adaptWalkProb = 0, phi=10, selectClause = 1, heuristic = 7, rdp = 0.15, theta = 3 singleClauseAsNeighbor = 0, performSearchDiversification = 0, useAdaptiveMechanism = 0, usePromisingList = 0, FAC useClausePenalty = 1, performTabuSearch = 0, scoringMeasure = 3, selectClause = 1, wp = 0, alpha = 1.126, smoothingScheme = 1, tieBreaking = 1, ps = 0.033 singleClauseAsNeighbor = 0, performSearchDiversification = 1, useAdaptiveMechanism = 1, searchDiversificationStrategy = 1, CBMC(SE) usePromisingList = 0, performTabuSearch = 0, useClausePenalty = 1, smoothingScheme = 1, selectClause = 1, tieBreaking = 1, wp = 0.01, rwp=0.01, scoringMeasure = 1, rho = 1, alpha = 1.126, singleClauseAsNeighbor = 1, performSearchDiversification = 1, useAdaptiveMechanism = 1, searchDiversificationStrategy = 4, usePromisingList = 1, performTabuSearch = 1, heuristic = 7 Random useClausePenalty = 0, updateSchemePromList = 1, theta = 4, selectPromVariable = 5, adaptiveNoiseScheme = 2, rfp = 0.07, adaptWalkProb = 0, tabuLength = 3, tieBreaking = 3, phi = 7, selectClause = 1 singleClauseAsNeighbor = 1, performSearchDiversification = 0, useAdaptiveMechanism = 0, usePromisingList = 1, heuristic = 6, Handmade useClausePenalty = 0, performTabuSearch 0, novNoise = 0.1, selectPromVariable = 10, updateSchemePromList = 2, promNovNoise = 0.6, selectClause = 1, adaptiveProm = 0 promDp = 0.01 singleClauseAsNeighbor = 0, performSearchDiversification = 0, useAdaptiveMechanism = 0, usePromisingList = 0, alpha = 1.126, Industrial performTabuSearch = 0, smoothingScheme = 1, tieBreaking = 3, useClausePenalty = 1, scoringMeasure = 1, ps = 0, rho = 0.83, wp = 0.06 singleClauseAsNeighbor = 0, performSearchDiversification = 0, useAdaptiveMechanism = 0, performTabuSearch = 1, Big-mix useClausePenalty = 1, usePromisingList = 0, alpha = 1.126 tabuLength = 1, smoothingScheme = 1, scoringMeasure = 3, tieBreaking = 1, ps = 0.033, rho = 0.666, wp = 0.04 Table 5.9: SATenstein-LS parameter configuration found for each distribution. 45 Chapter 6 Conclusions and Future Work In this work, we have proposed a new way of designing heuristic algorithms that is based on (1) a framework that can flexibly combine components drawn from existing high-performance solvers, and (2) a powerful algorithm configuration tool for finding instantiations that perform well on given sets of instances. We demonstrated the effectiveness of our approach by automatically constructing high-performance stochastic local search solvers for SAT. We presented empirical evidence that our automatically constructed SAT solvers outperformed existing state-of-the-art solvers in several widely studied distributions of SAT instances. Source code and documentation for our SATenstein-LS framework are available online at http://www.cs.ubc.ca/labs/beta/Projects/SATenstein. We believe that SATenstein-LS is a useful piece of software. Its flexibility allows the user to configure it as a broad range of SLS solvers. No previous SLS solver has a design space as vast as that of SATenstein-LS. Also, it in- herits all the reporting features that UBCSAT provides. For these reasons, SATenstein-LS can be very useful in studying the behavior of different SLS al- gorithms. Secondly, SATenstein-LS can benefit the development of SLS solvers. Instead of trying out new ideas from scratch, one can extend it with new ideas and then, configure the modified SATenstein-LS using an automated algorithm configurator. An inclusion of the new strategy in the configurations thus found will suggest that the new strategy has promise. Also, the configurations will provide insights into designs that best complement the new strategy. Finally, SATenstein-LS can be used to evaluate the performance of a new SLS solver. We can easily create a wrapper-solver that either instantiates a SATenstein-LS solver or behaves like the new solver, depending on a categorical parameter. Then, we can configure the wrapper-solver on different distributions. For a given distribution, the instantiation of the wrapper-solver as the new solver essentially means that the configurator chose the new solver over trillions of solvers that can be instantiated using SATenstein-LS. In this way, we can eas- ily compare the performance of a new SLS solver against that of a rich space of solvers. Although we have already demonstrated strong results that improve the state of the art in solving several types of SAT instances, our framework can be further improved in the following ways. First, SATenstein-LS can be extended by including an optional preprocessing block and exposing parameters that specify Chapter 6. Conclusions and Future Work 46 the combination of preprocessors. Second, the current design of SATenstein-LS does not allow combinations of components from the G2WSAT architecture and dynamic local search algorithms. None of the high-performance algorithms we studied combined components from these two categories. However, our results show that dynamic local search algorithms and G2WSAT-based algorithms tend to perform well in different distributions. Thus extending SATenstein-LS to allow such combinations may lead to finding algorithms with more robust performance across different distributions. Third, at present, our categorical parameters take integer values and can specify only one option among the available ones. SATenstein-LS can be extended to allow probabilistic choice among multiple options. In similar vein to the ideas behind GSAT with random walk (Selman and Kautz, 1993) and behind the more general GLSM model of SLS algorithms (Hoos and Stützle, 2005), we can extend the idea of assigning non-zero execution- probabilities to the use of SATenstein-LS building blocks 2, 3, and 4. This increases the size of the already huge design space. As a result, we may be able to find algorithms demonstrating even better performance than our present SATenstein-LS solvers. Finally, other interesting lines of work concerned with enhancing the framework include exploring the combination of SATenstein-LS solvers trained on various types of SAT instances by means of an algorithm selection approach (see, e.g., Xu et al., 2008), and the investigation of algorithm configuration procedures other than ParamILS in the context of our approach. Like Dr. Frankenstein, we find our creations somewhat monstrous, recogniz- ing that our SATenstein-LS solvers do not embody the most elegant designs. We have tried to relate SATenstein-LS solvers to known algorithms by present- ing a novel distance metric based on hierarchically arranged parameters. We have also described SATenstein-LS solvers in terms of known algorithm compo- nents. Nevertheless, additional work is required to gain a detailed understanding of how the SATenstein-LS solvers relate to previously known SAT algorithms and to explore which parameters are crucial for their superior performance. 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So, for a SAT instance ‘sample.cnf’, the simplest execution command for SATenstein-LS is > ubcsat -alg satenstein -i sample.cnf Once you execute this command, in the output, along with the run result (e.g., number of flips, runtime), you will notice a list of parameters. The execu- tion command you just performed ran SATenstein-LS with all the parameters set to their default values. You can find the default value of each parameter listed along with the parameter name. Similar to UBCSAT, the values for these parameters can be specified by [-paramname value]. For example, the execution command for running SATenstein-LS with the parameter adaptive set to 0 will be > ubcsat -alg satenstein -i sample.cnf -adaptive 0 A.3 Configurable Parameters Table 3.4 and 4.2 respectively present the continuous and integer parameters of SATenstein-LS, along with a short description of each parameter. The cate- gorical parameters are summarized in Table 3.2. Table 4.3 and 3.3 respectively present the parameter values we considered in our experiments. 1http://www.satlib.org/ubcsat/quickstart.html Appendix A. SATenstein-LS User Manual 55 The parameters that we expose on the command line have slightly different names from those we have used so far. Table A.1 shows the name mapping between SATenstein-LS parameters discussed so far and SATenstein-LS pa- rameters exposed on the command line. Parameters that do not require any such mapping are omitted from this Table. Parameter Name Command-Line Name singleClauseAsNeighbor singleclause usePromisingList promisingList useClausePenalty clausepen performTabuSearch tabusearch performSearchDiversification performrandomwalk useAdaptiveMechanism adaptive adaptWalkProb adaptwalkprob adaptiveProm adaptiveprom adaptPromWalkProb adaptPromWalkProb updateSchemePromList updateschemepromlist adaptiveNoiseScheme adaptivenoisescheme smoothingScheme smoothingscheme searchDiversificationStrategy randomwalk selectPromVariable decreasingvariable tieBreaking tiebreaking scoringMeasure scoringmeasure performAlternateNovelty performalternatenovelty selectClause selectclause promTheta promtheta promPhi promPhi promDp promdp promWp promwp promNovNoise promnovnoise novNoise novnoise wpWalk wpwalk psWalk ps Table A.1: Name mapping between SATenstein-LS parameters discussed so far and SATenstein-LS parameters exposed on the command line. A.4 Example Parameter Configurations Now we may configure some known algorithms with SATenstein-LS. Here we give brief outlines for the algorithms. For detailed description of the algorithms, please refer to the citepd papers. A.4.1 Novelty+ Let us start with a simple algorithm, Novelty+ (Hoos, 1999). This algorithm uses block 3 (-singleclause 1) (see Section 3.1) and does not use promising list -promisinglist 0). Table 3.6 shows that Novelty+ has two heuristic- specific parameters, novnoise and wp, and they have default values of 0.5 and 0.01, respectively. Table 3.6 also shows that heuristic needs to set to 2. Novelty+ does not use any adaptive mechanism (-adaptive 0). Hence, the command to configure SATenstein-LS as Novelty+ is Appendix A. SATenstein-LS User Manual 56 > ubcsat -alg satenstein -i sample.cnf -selectclause 1 -promisinglist 0 -adaptive 0 -heuristic 2 -wp 0.01 -novnoise 0.5 You may want to have a look at the default values of these parameters; you will find that the default values for selectclause, novnoise and wp are the same as the ones used to run Novelty+. Hence, this command can be simplified as > ubcsat -alg satenstein -i sample.cnf -promisinglist 0 -adaptive 0 -heuristic 2 Simple, isn’t it? You may want to run adaptNovelty+ (Hoos, 2002). For that, all that you need to do is to change the value of adaptive. The command to configure SATenstein-LS as adaptNovelty+ is: > ubcsat -alg satenstein -i sample.cnf -promisinglist 0 -adaptive 1 -heuristic 2 A.4.2 gNovelty+ gNovelty+ (Pham et al., 2008) uses block 2 (-promisinglist 1) (see Sec- tion 3.1), block 1 (-performrandomwalk 1), clause penalties (-clausepen 1), smoothing (-smoothingscheme 1), and adaptive mechanisms (-adaptive 1 -adaptwalkprob 0). It picks the variable with highest score from a non-empty promising list (-decreasingvariable 2). If the list is empty, it performs the Novelty heuristic (with “flat moves”) (-heuristic 1 -performalternatenovelty 1). In module 1, gNovelty+ randomly picks a variable (-randomwalk 1) from a randomly selected false clause ( -selectclause 1) with a probability 0.01 (-rwp 0.01). The command to configure SATenstein-LS as gNovelty+ is: > ubcsat -alg satenstein -inst sample.cnf -selectclause 1 -performrandomwalk 1 -randomwalk 1 -rwp 0.01 -promisinglist 1 -decreasingvariable 2 -updateschemepromlist 3 -adaptive 1 -adaptwalkprob 0 -adaptivenoisecheme 1 -heuristic 1 -smoothingscheme 1 -clausepen 1 -smoothingscheme 1 As explained earlier, we can simplify this command by eliminating parame- ters whose values are same as their respective default values. A.4.3 SAPS SAPS (Hutter et al., 2002) uses block 4 (see Section 3.1) (-promisinglist 0 -singleclause 0) and does not use any adaptive mechanism (-adaptive 0). (Makecount - Breakcount) is used to score a variable (-scoringmeasure 1) and any remaining ties are broken randomly (-tiebreaking 1). It uses clause Appendix A. SATenstein-LS User Manual 57 penalties (-clausepen 1) and multiplicative smoothing scheme (-smoothingscheme 1). Clearly, the command for running SAPS with the SAPS-specific parameters alpha, rho, and wp set to their respective default values, is > ubcsat -alg satenstein -i sample.cnf -singleclause 0 -promisinglist 0 -scoringmeasure 1 -tiebreaking 1 -clausepen 1 -smoothingscheme 1 -adaptive 0 Since RSAPS (Hutter et al., 2002) is SAPS with a reactive mechanism, by tog- gling the value of adaptive in the above command, we can configure SATenstein-LS as RSAPS. PAWS uses exactly the same configuration as SAPS except for the smoothing scheme. By changing the value of smoothingscheme to 2 in the above command, we can configure SATenstein-LS as PAWS. The two PAWS specific parameters maxinc and pflat have default values of 10 and 0.15, respectively.

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