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DOI: 10.4230/LIPIcs.SEA.2025.27

Sparsity-Driven Aggregation of Mixed Integer Programs

Authors: Liding Xu, Gioni Mexi, and Ksenia Bestuzheva

Published in: LIPIcs, Volume 338, 23rd International Symposium on Experimental Algorithms (SEA 2025)

Abstract

Cutting planes are crucial for the performance of branch-and-cut algorithms for solving mixed-integer programming (MIP) problems, and linear row aggregation has been successfully applied to better leverage the potential of several major families of MIP cutting planes. This paper formulates the problem of finding good quality aggregations as an 𝓁₀-norm minimization problem and employs a combination of the lasso method and iterative reweighting to efficiently find sparse solutions corresponding to good aggregations. A comparative analysis of the proposed algorithm and the state-of-the-art greedy heuristic approach is presented, showing that the greedy heuristic implements a stepwise selection algorithm for the 𝓁₀-norm minimization problem. Further, we present an example where our approach succeeds, whereas the standard heuristic fails to find an aggregation with desired properties. The algorithm is implemented within the constraint integer programming solver SCIP, and computational experiments on the MIPLIB 2017 benchmark show that although the algorithm leads to slowdowns on relatively "easier" instances, our aggregation approach decreases the mean running time on a subset of challenging instances and leads to smaller branch-and-bound trees.

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Liding Xu, Gioni Mexi, and Ksenia Bestuzheva. Sparsity-Driven Aggregation of Mixed Integer Programs. In 23rd International Symposium on Experimental Algorithms (SEA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 338, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{xu_et_al:LIPIcs.SEA.2025.27,
  author =	{Xu, Liding and Mexi, Gioni and Bestuzheva, Ksenia},
  title =	{{Sparsity-Driven Aggregation of Mixed Integer Programs}},
  booktitle =	{23rd International Symposium on Experimental Algorithms (SEA 2025)},
  pages =	{27:1--27:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-375-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{338},
  editor =	{Mutzel, Petra and Prezza, Nicola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2025.27},
  URN =		{urn:nbn:de:0030-drops-232652},
  doi =		{10.4230/LIPIcs.SEA.2025.27},
  annote =	{Keywords: mixed integer linear programming, cutting plane, valid inequality, separation, aggregation, projection, sparse optimization}
}

Document

DOI: 10.4230/LIPIcs.CP.2023.27

Improving Conflict Analysis in MIP Solvers by Pseudo-Boolean Reasoning

Authors: Gioni Mexi, Timo Berthold, Ambros Gleixner, and Jakob Nordström

Published in: LIPIcs, Volume 280, 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 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 MIPLIB2017, 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.

Cite as

Gioni Mexi, Timo Berthold, Ambros Gleixner, and Jakob Nordström. Improving Conflict Analysis in MIP Solvers by Pseudo-Boolean Reasoning. In 29th International Conference on Principles and Practice of Constraint Programming (CP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 280, pp. 27:1-27:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{mexi_et_al:LIPIcs.CP.2023.27,
  author =	{Mexi, Gioni and Berthold, Timo and Gleixner, Ambros and Nordstr\"{o}m, Jakob},
  title =	{{Improving Conflict Analysis in MIP Solvers by Pseudo-Boolean Reasoning}},
  booktitle =	{29th International Conference on Principles and Practice of Constraint Programming (CP 2023)},
  pages =	{27:1--27:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-300-3},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{280},
  editor =	{Yap, Roland H. C.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2023.27},
  URN =		{urn:nbn:de:0030-drops-190641},
  doi =		{10.4230/LIPIcs.CP.2023.27},
  annote =	{Keywords: Integer programming, pseudo-Boolean solving, conflict analysis, cutting planes proof system, mixed integer rounding, division, saturation}
}

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Sparsity-Driven Aggregation of Mixed Integer Programs

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Improving Conflict Analysis in MIP Solvers by Pseudo-Boolean Reasoning

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