An Unconstrained Quadratic Binary Programming Approach to the Vertex Coloring Problem

Abstract

The vertex coloring problem has been the subject of extensive research for many years. Driven by application potential as well as computational challenge, a variety of methods have been proposed for this difficult class of problems. Recent successes in the use of the unconstrained quadratic programming (UQP) model as a unified framework for modeling and solving combinatorial optimization problems have motivated a new approach to the vertex coloring problem. In this paper we present a UQP approach to this problem and illustrate its attractiveness with preliminary computational experience.

Solving the maximum edge weight clique problem via unconstrained quadratic programming

“…As noted in recent papers (see for instance (Alidaee et al, 2005(Alidaee et al, , 2006Kochenberger et al, 2005aKochenberger et al, , 2004aKochenberger et al, ,b, 2005b, the model UQP has proven to function efficiently and effectively as a unified framework for modeling and solving a wide variety of combinatorial optimization problems. In the context of other problem classes, we have solved instances of UQP with more than 50,000 variables, which means we could conceivably solve the nonlinear version of MEWCP for graphs with more than 50,000 nodes.…”

A hybrid metaheuristic approach to solving the UBQP problem

“…This re-formulation process enables UBQP to serve as a common model for a wide range of combinatorial optimization problems. A review of additional applications and the re-formulation procedures can be found in [26] demonstrating the utility of UBQP for a variety of applications, such as the vertex coloring problem [27], the set packing problem [2], the set-partitioning problem [30] and the linear ordering problem [31].…”

Diversification-driven tabu search for unconstrained binary quadratic problems

“…Moreover, many combinatorial optimization problems pertaining to graphs such as determining maximum cliques, maximum cuts, maximum vertex packing, minimum coverings, maximum independent sets, maximum independent weighted sets are known to be capable of being formulated by the UBQP problem as documented in papers of Pardalos and Rodgers (1990), Pardalos and Xue (1994). A review of additional applications and formulations can be found in Kochenberger et al (2004Kochenberger et al ( , 2005, Alidaee et al (2008), Lewis et al (2008).…”