The Hirsch Conjecture for the fractional stable set polytope

• 成果类型: Article
• 署名作者: Michini, Carla; Sassano, Antonio
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich; Sapienza University Rome
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-013-0723-3
• 发表日期: 2014
• 页码: 309-330
• 关键词: polyhedra

摘要

The edge formulation of the stable set problem is defined by two-variable constraints, one for each edge of a graph , expressing the simple condition that two adjacent nodes cannot belong to a stable set. We study the fractional stable set polytope, i.e. the polytope defined by the linear relaxation of the edge formulation. Even if this polytope is a weak approximation of the stable set polytope, its simple geometrical structure provides deep theoretical insight as well as interesting algorithmic opportunities. Exploiting a graphic characterization of the bases, we first redefine pivots in terms of simple graphic operations, that turn a given basis into an adjacent one. These results lead us to prove that the combinatorial diameter of the fractional stable set polytope is at most the number of nodes of the given graph. As a corollary, the Hirsch bound holds for this class of polytopes.