Strengthened Partial-Ordering Based ILP Models for the Vertex Coloring Problem

TL;DR

This paper enhances partial-ordering based ILP models for the vertex coloring problem by introducing new constraints, resulting in improved bounds and the ability to solve previously open benchmark instances.

Contribution

The paper proposes additional strengthening constraints for partial-ordering ILP models, improving their effectiveness in solving the vertex coloring problem.

Findings

Improved lower bounds from strengthened ILP models

Practical improvements in solving benchmark instances

Solved a previously open DIMACS benchmark instance

Abstract

The vertex coloring problem asks for the minimum number of colors that can be assigned to the vertices of a given graph such that each two adjacent vertices get different colors. For this NP-hard problem, a variety of integer linear programming (ILP) models have been suggested. Among them, the assignment based and the partial-ordering based ILP models are attractive due to their simplicity and easy extendability. Moreover, on sparse graphs, both models turned out to be among the strongest exact approaches for solving the vertex coloring problem. In this work, we suggest additional strengthening constraints for the partial-ordering based ILP model, and show that they lead to improved lower bounds of the corresponding LP relaxation. Our computational experiments confirm that the new constraints are also leading to practical improvements. In particular, we are able to find the optimal solution of a previously open instance from the DIMACS benchmark set for vertex coloring problems