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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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Average Case Polyhedral Complexity of the Maximum Stable Set Problem

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