Upper and lower bounds based on linear programming for the b-coloring problem

Abstract

B-coloring is a problem in graph theory. It can model some real applications, as well as being used to enhance solution methods for the classical graph coloring problem. In turn, improved solutions for the classical coloring problem would impact a larger pool of practical applications in several different fields such as scheduling, timetabling and telecommunications. Given a graph $G=(V,E)$, the b-coloring problem aims to maximize the number of colors used while assigning a color to every vertex in V, preventing adjacent vertices from receiving the same color, with every color represented by a special vertex, called a b-vertex. A vertex can be a b-vertex only if the set of colors assigned to its adjacent vertices includes all the colors, apart from the one assigned to the vertex itself.

This work employs methods based on Linear Programming to derive new upper and lower bounds for the problem. In particular, starting from a Mixed Integer Linear Programming model recently presented, upper bounds are obtained through partial linear relaxations of this model, while lower bounds are derived by considering different variations of the original model, modified to target a specific number of colors provided as input. The experimental campaign documented in the paper led to several improvements to the state-of-the-art results.