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Extension Complexity, MSO Logic, and Treewidth

Authors Petr Kolman, Martin Koutecký, Hans Raj Tiwary

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Petr Kolman
Martin Koutecký
Hans Raj Tiwary

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Petr Kolman, Martin Koutecký, and Hans Raj Tiwary. Extension Complexity, MSO Logic, and Treewidth. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.SWAT.2016.18

Abstract

We consider the convex hull P_phi(G) of all satisfying assignments of a given MSO_2 formula phi on a given graph G. We show that there exists an extended formulation of the polytope P_phi(G) that can be described by f(|phi|,tau)*n inequalities, where n is the number of vertices in G, tau is the treewidth of G and f is a computable function depending only on phi and tau.

In other words, we prove that the extension complexity of P_phi(G) is linear in the size of the graph G, with a constant depending on the treewidth of G and the formula phi.  This provides a very general yet very simple meta-theorem about the extension complexity of polytopes related to a wide class of problems and graphs.

Subject Classification

Keywords
  • Extension Complexity
  • FPT
  • Courcelle's Theorem
  • MSO Logic

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