Stable Sets and Graphs with no Even Holes

TL;DR

This paper introduces decomposition tools for solving maximum weight stable set problems and describes them as polynomial linear programs, specifically applied to graphs without even holes or caps, enhancing algorithmic efficiency.

Contribution

It develops polynomial decomposition schemes and linear programming descriptions for stable sets in graphs with no even holes or caps, advancing graph optimization methods.

Findings

Decomposition schemes for stable sets are polynomial.

Linear programming formulations are compact and efficient.

Applicable to graphs with no even holes or caps.

Abstract

We develop decomposition/composition tools for efficiently solving maximum weight stable sets problems as well as for describing them as polynomially sized linear programs (using "compact systems"). Some of these are well-known but need some extra work to yield polynomial "decomposition schemes". We apply the tools to graphs with no even hole and no cap. A hole is a chordless cycle of length greater than three and a cap is a hole together with an additional node that is adjacent to two adjacent nodes of the hole and that has no other neighbors on the hole.