Convex Recoloring of General Graphs: Formulations, Polyhedra, and Computational Experiments

2026-06-29

AI Summary

This study addresses the convex recoloring problem on general graphs, which seeks to minimize the total weight of recolored vertices so that the subgraph induced by each color class is connected. The work presents the first systematic formulation of four mixed-integer linear programming models, including a novel compact flow-based model and a representative-vertex model, and provides a thorough polyhedral analysis of their linear relaxations. Leveraging insights from polyhedral theory and graph-theoretic modeling, the authors design and implement corresponding branch-and-cut algorithms. Computational experiments demonstrate that the branch-and-cut algorithm based on the representative-vertex model achieves superior performance on both standard and synthetic benchmark instances, significantly advancing the state of the art in exact solution methods for this problem.

Abstract

A vertex coloring of a graph is convex if the vertices of each color induce a connected subgraph. In the convex recoloring problem (CR), the goal is to find a convex coloring while minimizing the weight of recolored vertices, i.e., vertices assigned a color different from their original one. This problem was originally motivated by the study of phylogenetic trees in bioinformatics and is NP-hard even on paths. Most existing research focuses on trees, with only limited results available for general graphs. We advance the state of the art by developing exact solution methods for CR on general graphs. In particular, we propose four mixed-integer linear programming formulations, including a compact flow-based model and a representatives model, and design corresponding solution methods. We compare the polytopes associated with the linear relaxation of the proposed formulations. Computational experiments on benchmark instances and on new synthetic instances show that a branch-and-cut algorithm based on the representatives formulation performs best overall.

Problem

• convex recoloring
• general graphs
• vertex coloring
• NP-hard
• connected subgraph

Innovation

• convex recoloring
• mixed-integer linear programming
• polyhedral comparison
• branch-and-cut
• general graphs