Extended Formulations for Stable Set Polytopes of Graphs Without Two Disjoint Odd Cycles

TL;DR

This paper constructs a compact extended formulation for the stable set polytope of graphs without two disjoint odd cycles, enabling efficient optimization in such graphs by leveraging structural properties.

Contribution

It provides a size-$O(n^2)$ extended formulation for the stable set polytope of these graphs, advancing understanding of their polyhedral structure.

Findings

Constructed a quadratic-size extended formulation for the stable set polytope.

Enabled polynomial-time optimization for stable sets in these graphs.

Linked structural graph properties to polyhedral representations.

Abstract

Let $G$ be an $n$-node graph without two disjoint odd cycles. The algorithm of Artmann, Weismantel and Zenklusen (STOC'17) for bimodular integer programs can be used to find a maximum weight stable set in $G$ in strongly polynomial time. Building on structural results characterizing sufficiently connected graphs without two disjoint odd cycles, we construct a size-$O(n^2)$ extended formulation for the stable set polytope of $G$.