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

2025-12-08

AI Summary

This paper studies the Maximum Weighted co-2-plex finding a maximum-weight vertex subset in a weighted graph such that its induced subgraph has maximum degree at most one. We establish, for the first time, a bijection between feasible co-2-plexes and stable sets in an auxiliary graph, embed the co-2-plex polytope into an extended space, and characterize input-graph contraction-perfectness via total dual integrality. Based on this structural insight, we derive a family of valid inequalities separable in polynomial time, uncovering deep connections between the extended formulation and the integer polytope. The resulting integer linear programming (ILP) model yields tighter linear programming relaxations. Experiments demonstrate that our approach significantly outperforms state-of-the-art methods across sparse to dense graphs, achieving substantial improvements in computational efficiency.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Characterizes co-2-plex polytopes for contraction perfect graphs using extended formulations.
• Projects extended formulations to obtain facet-defining inequalities for trees.
• Develops tighter integer linear programs for maximum weighted co-2-plex optimization.

Innovation

• Extended polyhedral formulations using auxiliary graph bijection
• Projection yields new valid inequalities with facet characterization
• Polynomial-time separable inequalities for tighter ILP relaxations