Polyhedral study of the Convex Recoloring problem

TL;DR

This paper studies the convex recoloring problem in graphs, introducing a polyhedral framework, valid inequalities, and computational experiments to improve understanding and solution approaches for this NP-hard problem.

Contribution

It introduces a polyhedral model for convex recoloring, identifies new valid inequalities, and evaluates their effectiveness through computational experiments.

Findings

Polyhedral model captures convex recoloring constraints.

New valid inequalities improve LP relaxation bounds.

Computational results show potential for reducing integrality gaps.

Abstract

A coloring of the vertices of a connected graph is convex if each color class induces a connected subgraph. We address the convex recoloring (CR) problem defined as follows. Given a graph $G$ and a coloring of its vertices, recolor a minimum number of vertices of $G$ so that the resulting coloring is convex. This problem, known to be NP-hard even on paths, was firstly motivated by applications on perfect phylogenies. In this work, we study CR on general graphs from a polyhedral point of view. First, we introduce a full-dimensional polytope based on the idea of connected subgraphs, and present a class of valid inequalities with righthand side one that comprises all facet-defining inequalities with binary coefficients when the input graph is a tree. Moreover, we define a general class of inequalities with righthand side in $\{1, \ldots, k\}$, where $k$ is the amount of colors used in the initial coloring, and show sufficient conditions for validity and facetness of such inequalities. Finally, we report on computational experiments for an application on mobile networks that can be modeled by the polytope of CR on paths. We evaluate the potential of the proposed inequalities to reduce the integrality gaps.