A column generation algorithm for finding co-3-plexes in chordal graphs

TL;DR

This paper presents a polynomial-time column generation algorithm for finding maximum co-3-plexes in chordal graphs by reducing the problem to a maximum stable set problem in an auxiliary graph.

Contribution

It introduces a novel exponential LP formulation and a polynomial-time solution method for maximum co-3-plexes specifically in chordal graphs.

Findings

The problem reduces to finding a maximum stable set in an auxiliary graph.

The pricing subproblem simplifies to a maximum vertex and edge weighted induced path.

The algorithm is polynomial-time for chordal graphs.

Abstract

In this study, we tackle the problem of finding a maximum \emph{co-3-plex}, which is a subset of vertices of an input graph, inducing a subgraph of maximum degree 2. We focus on the class of chordal graphs. By observing that the graph induced by a co-3-plex in a chordal graph is a set of isolated triangles and induced paths, we reduce the problem of finding a maximum weight co-3-plex in a graph $G$ to that of finding a maximum stable set in an auxiliary graph $\mathcal{A}(G)$ of exponential size. This reduction allows us to derive an exponential variable-sized linear programming formulation for the maximum weighted co-3-plex problem. We show that the pricing subproblem of this formulation reduces to solving a maximum vertex and edge weight induced path. Such a problem is solvable in polynomial time; therefore, this exhibits a polynomial time column generation algorithm solving the maximum co-3-plex problem on chordal graphs. Moreover, this machinery exhibits a new application for the maximum vertex and edge weighted induced path problems.