Improving Conflict Analysis in MIP Solvers by Pseudo-Boolean Reasoning
(2023) 29th International Conference on Principles and Practice of Constraint Programming, CP 2023- Abstract
Conflict analysis has been successfully generalized from Boolean satisfiability (SAT) solving to mixed integer programming (MIP) solvers, but although MIP solvers operate with general linear inequalities, the conflict analysis in MIP has been limited to reasoning with the more restricted class of clausal constraint. This is in contrast to how conflict analysis is performed in so-called pseudo-Boolean solving, where solvers can reason directly with 0-1 integer linear inequalities rather than with clausal constraints extracted from such inequalities. In this work, we investigate how pseudo-Boolean conflict analysis can be integrated in MIP solving, focusing on 0-1 integer linear programs (0-1 ILPs). Phrased in MIP terminology, conflict... (More)
Conflict analysis has been successfully generalized from Boolean satisfiability (SAT) solving to mixed integer programming (MIP) solvers, but although MIP solvers operate with general linear inequalities, the conflict analysis in MIP has been limited to reasoning with the more restricted class of clausal constraint. This is in contrast to how conflict analysis is performed in so-called pseudo-Boolean solving, where solvers can reason directly with 0-1 integer linear inequalities rather than with clausal constraints extracted from such inequalities. In this work, we investigate how pseudo-Boolean conflict analysis can be integrated in MIP solving, focusing on 0-1 integer linear programs (0-1 ILPs). Phrased in MIP terminology, conflict analysis can be understood as a sequence of linear combinations and cuts. We leverage this perspective to design a new conflict analysis algorithm based on mixed integer rounding (MIR) cuts, which theoretically dominates the state-of-the-art division-based method in pseudo-Boolean solving. We also report results from a first proof-of-concept implementation of different pseudo-Boolean conflict analysis methods in the open-source MIP solver SCIP. When evaluated on a large and diverse set of 0-1 ILP instances from MIPLIB 2017, our new MIR-based conflict analysis outperforms both previous pseudo-Boolean methods and the clause-based method used in MIP. Our conclusion is that pseudo-Boolean conflict analysis in MIP is a promising research direction that merits further study, and that it might also make sense to investigate the use of such conflict analysis to generate stronger no-goods in constraint programming.
(Less)
- author
- Mexi, Gioni ; Berthold, Timo ; Gleixner, Ambros and Nordström, Jakob LU
- organization
- publishing date
- 2023-09
- type
- Chapter in Book/Report/Conference proceeding
- publication status
- published
- subject
- keywords
- conflict analysis, cutting planes proof system, division, Integer programming, mixed integer rounding, pseudo-Boolean solving, saturation
- host publication
- 29th International Conference on Principles and Practice of Constraint Programming, CP 2023
- editor
- Yap, Roland H. C. Yap
- article number
- 27
- publisher
- Schloss Dagstuhl - Leibniz-Zentrum für Informatik
- conference name
- 29th International Conference on Principles and Practice of Constraint Programming, CP 2023
- conference location
- Toronto, Canada
- conference dates
- 2023-08-27 - 2023-08-31
- external identifiers
-
- scopus:85174068835
- ISBN
- 9783959773003
- DOI
- 10.4230/LIPIcs.CP.2023.27
- language
- English
- LU publication?
- yes
- id
- 96d51e2e-1266-4950-a767-73ab52d53e47
- date added to LUP
- 2023-12-11 14:52:52
- date last changed
- 2024-02-09 11:28:14
@inproceedings{96d51e2e-1266-4950-a767-73ab52d53e47,
abstract = {{<p>Conflict analysis has been successfully generalized from Boolean satisfiability (SAT) solving to mixed integer programming (MIP) solvers, but although MIP solvers operate with general linear inequalities, the conflict analysis in MIP has been limited to reasoning with the more restricted class of clausal constraint. This is in contrast to how conflict analysis is performed in so-called pseudo-Boolean solving, where solvers can reason directly with 0-1 integer linear inequalities rather than with clausal constraints extracted from such inequalities. In this work, we investigate how pseudo-Boolean conflict analysis can be integrated in MIP solving, focusing on 0-1 integer linear programs (0-1 ILPs). Phrased in MIP terminology, conflict analysis can be understood as a sequence of linear combinations and cuts. We leverage this perspective to design a new conflict analysis algorithm based on mixed integer rounding (MIR) cuts, which theoretically dominates the state-of-the-art division-based method in pseudo-Boolean solving. We also report results from a first proof-of-concept implementation of different pseudo-Boolean conflict analysis methods in the open-source MIP solver SCIP. When evaluated on a large and diverse set of 0-1 ILP instances from MIPLIB 2017, our new MIR-based conflict analysis outperforms both previous pseudo-Boolean methods and the clause-based method used in MIP. Our conclusion is that pseudo-Boolean conflict analysis in MIP is a promising research direction that merits further study, and that it might also make sense to investigate the use of such conflict analysis to generate stronger no-goods in constraint programming.</p>}},
author = {{Mexi, Gioni and Berthold, Timo and Gleixner, Ambros and Nordström, Jakob}},
booktitle = {{29th International Conference on Principles and Practice of Constraint Programming, CP 2023}},
editor = {{Yap, Roland H. C. Yap}},
isbn = {{9783959773003}},
keywords = {{conflict analysis; cutting planes proof system; division; Integer programming; mixed integer rounding; pseudo-Boolean solving; saturation}},
language = {{eng}},
publisher = {{Schloss Dagstuhl - Leibniz-Zentrum für Informatik}},
title = {{Improving Conflict Analysis in MIP Solvers by Pseudo-Boolean Reasoning}},
url = {{http://dx.doi.org/10.4230/LIPIcs.CP.2023.27}},
doi = {{10.4230/LIPIcs.CP.2023.27}},
year = {{2023}},
}