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DOI: 10.4230/LIPIcs.CP.2021.52

An Algorithm-Independent Measure of Progress for Linear Constraint Propagation

Authors: Boro Sofranac, Ambros Gleixner, and Sebastian Pokutta

Published in: LIPIcs, Volume 210, 27th International Conference on Principles and Practice of Constraint Programming (CP 2021)

Abstract

Propagation of linear constraints has become a crucial sub-routine in modern Mixed-Integer Programming (MIP) solvers. In practice, iterative algorithms with tolerance-based stopping criteria are used to avoid problems with slow or infinite convergence. However, these heuristic stopping criteria can pose difficulties for fairly comparing the efficiency of different implementations of iterative propagation algorithms in a real-world setting. Most significantly, the presence of unbounded variable domains in the problem formulation makes it difficult to quantify the relative size of reductions performed on them. In this work, we develop a method to measure - independently of the algorithmic design - the progress that a given iterative propagation procedure has made at a given point in time during its execution. Our measure makes it possible to study and better compare the behavior of bounds propagation algorithms for linear constraints. We apply the new measure to answer two questions of practical relevance: (i) We investigate to what extent heuristic stopping criteria can lead to premature termination on real-world MIP instances. (ii) We compare a GPU-parallel propagation algorithm against a sequential state-of-the-art implementation and show that the parallel version is even more competitive in a real-world setting than originally reported.

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Boro Sofranac, Ambros Gleixner, and Sebastian Pokutta. An Algorithm-Independent Measure of Progress for Linear Constraint Propagation. In 27th International Conference on Principles and Practice of Constraint Programming (CP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 210, pp. 52:1-52:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{sofranac_et_al:LIPIcs.CP.2021.52,
  author =	{Sofranac, Boro and Gleixner, Ambros and Pokutta, Sebastian},
  title =	{{An Algorithm-Independent Measure of Progress for Linear Constraint Propagation}},
  booktitle =	{27th International Conference on Principles and Practice of Constraint Programming (CP 2021)},
  pages =	{52:1--52:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-211-2},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{210},
  editor =	{Michel, Laurent D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2021.52},
  URN =		{urn:nbn:de:0030-drops-153430},
  doi =		{10.4230/LIPIcs.CP.2021.52},
  annote =	{Keywords: Bounds Propagation, Mixed Integer Programming}
}

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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.515

Average Case Polyhedral Complexity of the Maximum Stable Set Problem

Authors: Gábor Braun, Samuel Fiorini, and Sebastian Pokutta

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Abstract

We study the minimum number of constraints needed to formulate random instances of the maximum stable set problem via LPs (more precisely, linear extended formulations), in two distinct models. In the uniform model, the constraints of the LP are not allowed to depend on the input graph, which should be encoded solely in the objective function. There we prove a super-polynomial lower bound with overwhelming probability for every LP that is exact for a randomly selected set of instances with a natural distribution. In the non-uniform model, the constraints of the LP may depend on the input graph, but we allow weights on the vertices. The input graph is sampled according to the Erdös-Renyi model. There we obtain upper and lower bounds holding with high probability for various ranges of p. We obtain a super-polynomial lower bound all the way from essentially p = polylog(n) / n to p = 1 / log n. Our upper bound is close as there is only an essentially quadratic gap in the exponent, which also exists in the worst case model. Finally, we state a conjecture to close the gap both in the average-case and worst-case models.

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Gábor Braun, Samuel Fiorini, and Sebastian Pokutta. Average Case Polyhedral Complexity of the Maximum Stable Set Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 515-530, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{braun_et_al:LIPIcs.APPROX-RANDOM.2014.515,
  author =	{Braun, G\'{a}bor and Fiorini, Samuel and Pokutta, Sebastian},
  title =	{{Average Case Polyhedral Complexity of the Maximum Stable Set Problem}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{515--530},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.515},
  URN =		{urn:nbn:de:0030-drops-47201},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.515},
  annote =	{Keywords: polyhedral approximation, extended formulation, stable sets}
}

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An Algorithm-Independent Measure of Progress for Linear Constraint Propagation

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Average Case Polyhedral Complexity of the Maximum Stable Set Problem

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