Extended formulations for the maximum weighted co-2-plex problem
Alexandre Dupont-Bouillard, Pierre Fouilhoux, Roland Grappe, Mathieu Lacroix
TL;DR
The paper develops extended space polyhedral formulations for the maximum weighted co-2-plex problem by leveraging the stable-set polytope of an auxiliary utter graph. It establishes a tight link between contraction perfect graphs and integrality/TDI properties of the associated linear systems, and provides a compact extended description for chordal graphs. Projecting these extended formulations yields a natural-space description for trees and a new family of facet-defining inequalities for co-2-plex polytopes, including a polynomial-time separation for star inequalities. The authors introduce tighter ILPs (E(G)) that outperform the existing N_2(G) relaxations on some instances, and provide an extensive computational study showing that, while the extended formulations improve LP tightness, the classic formulations remain the faster choice on many datasets, particularly sparse graphs. Overall, the work advances both the theory (through contraction-perfectness and TDI characterizations) and practice (via stronger ILPs and separations) for maximum co-2-plex optimization.
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.