Extended formulations for the maximum weighted co-2-plex problem

TL;DR

This paper develops extended formulations and valid inequalities for the maximum weighted co-2-plex problem, providing tighter ILP relaxations and polynomial-time separation algorithms, with experimental validation on graph instances.

Contribution

It introduces new extended space formulations and valid inequalities for co-2-plex polytopes, improving ILP relaxation tightness and computational performance.

Findings

New extended formulations for co-2-plex polytopes.

Polynomial-time separation algorithms for new inequalities.

Tighter ILP relaxations outperforming the state of the art.

Abstract

Given an input graph and weights on its vertices, the maximum co-2-plex problem is to find a subset of vertices maximizing the sum of their weights and inducing a graph of degree at most 1. In this article, we analyze polyhedral aspects of the maximum co-2-plex problem. The co-2-plexes of a graph are known to be in bijection with the stable sets of an auxiliary graph called the utter graph~\cite{dupontbouillard2024contractions}. We use this bijection to characterize contraction perfect graphs' co-2-plex polytopes in an extended space. It turns out that the total dual integrality of the associated linear system also characterizes contraction perfectness of the input graph. By projecting this extended space formulation, we obtain the natural variable space description of the co-2-plex polytopes of trees. This projection yields a new family of valid inequalities for the co-2-plex polytope and we characterize when they define facets. Moreover, we show that these inequalities can be separated in polynomial time. We characterize the graphs for which this formulation describes an integer polytope. These linear systems are extended to valid integer linear programs (ILPs) for the maximum co-2-plex problem whose linear relaxation values are tighter than the state of the art for this problem~\cite{bala}. Finally, we provide an experimental comparison of several implementations of our new ILP formulations with the state-of-the-art ILP for this problem and analyze their respective performances relatively to the density of the input graphs.