Average case polyhedral complexity of the maximum stable set problem

详细信息    查看全文

• 作者：Gábor Braun ; Samuel Fiorini ; Sebastian Pokutta
• 关键词：Mathematics Subject ClassificationPrimary 68Q17 ; Secondary 05C69 ; 05C80 ; 90C05
• 刊名：Mathematical Programming
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：160
• 期：1-2
• 页码：407-431
• 全文大小：634 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Calculus of Variations and Optimal Control
  Mathematics of Computing
  Numerical Analysis
  Combinatorics
  Mathematical and Computational Physics
  Mathematical Methods in Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1436-4646
• 卷排序：160

文摘

We study the minimum number of constraints needed to formulate random instances of the maximum stable set problem via linear programs (LPs), 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 \(2^{\Omega (n{/}\log n)}\) lower bound with probability at least \(1 - 2^{-2^n}\) for every LP that is exact for a randomly selected set of instances; each graph on at most n vertices being selected independently with probability \(p \geqslant 2^{- \left( {\begin{array}{c}n/4\\ 2\end{array}}\right) + n}\). 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 G(n, p) 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 \(p = \Omega \left( \frac{\log ^{6+\varepsilon } n}{n} \right) \) to \(p = o\left( \frac{1}{\log n} \right) \). Our upper bound is close to this as there is only an essentially quadratic gap in the exponent, which currently also exists in the worst-case model. Finally, we state a conjecture that would close this gap, both in the average-case and worst-case models.