- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- BIRS Workshop Lecture Videos /
- Bimodular Integer Linear Programming and Beyond
Open Collections
BIRS Workshop Lecture Videos
BIRS Workshop Lecture Videos
Bimodular Integer Linear Programming and Beyond Zenklusen, Rico
Description
In this talk, I will show how any integer linear program (ILP) defined by a constraint matrix whose subdeterminants are all within {-2,-1,0,1,2} can be solved efficiently; even in strongly polynomial time. This is a natural extension of the well-known fact that ILPs with totally unimodular (TU) constraint matrices are polynomial-time solvable, which readily follows by the natural integrality of polytopes defined by a TU constraint matrix and integral right-hand sides. To derive this result we combine several techniques. In particular, the problem is first reduced to a particular parity-constrained ILP over a TU constraint matrix. We then leverage Seymour's decomposition of TU matrices to break this parity-constrained ILP into simpler base problems. Finally, we show how these base problems can be solved efficiently by combinatorial optimization techniques, including parity-constrained submodular minimization, and how to derive an optimal solution to the initial ILP from optimal solutions to base problems. Moreover, I will highlight some of the many open problems in this field and discuss recent results related to possible extensions to larger subdeterminants.
Item Metadata
Title |
Bimodular Integer Linear Programming and Beyond
|
Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
|
Date Issued |
2017-11-13T09:05
|
Description |
In this talk, I will show how any integer linear program (ILP) defined by a constraint matrix whose subdeterminants are all within {-2,-1,0,1,2} can be solved efficiently; even in strongly polynomial time. This is a natural extension of the well-known fact that ILPs with totally unimodular (TU) constraint matrices are polynomial-time solvable, which readily follows by the natural integrality of polytopes defined by a TU constraint matrix and integral right-hand sides.
To derive this result we combine several techniques. In particular, the problem is first reduced to a particular parity-constrained ILP over a TU constraint matrix. We then leverage Seymour's decomposition of TU matrices to break this parity-constrained ILP into simpler base problems. Finally, we show how these base problems can be solved efficiently by combinatorial optimization techniques, including parity-constrained submodular minimization, and how to derive an optimal solution to the initial ILP from optimal solutions to base problems. Moreover, I will highlight some of the many open problems in this field and discuss recent results related to possible extensions to larger subdeterminants.
|
Extent |
60 minutes
|
Subject | |
Type | |
File Format |
video/mp4
|
Language |
eng
|
Notes |
Author affiliation: ETH Zurich
|
Series | |
Date Available |
2018-05-13
|
Provider |
Vancouver : University of British Columbia Library
|
Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
|
DOI |
10.14288/1.0366260
|
URI | |
Affiliation | |
Peer Review Status |
Unreviewed
|
Scholarly Level |
Faculty
|
Rights URI | |
Aggregated Source Repository |
DSpace
|
Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International