A survey on vertex coloring problems

Abstract

This paper surveys the most important algorithmic and computational results on the Vertex Coloring Problem (VCP) and its generalizations. The first part of the paper introduces the classical models for the VCP, and discusses how these models can be used and possibly strengthened to derive exact and heuristic algorithms for the problem. Computational results on the best performing algorithms proposed in the literature are reported. The second part of the paper is devoted to some generalizations of the problem, which are obtained by considering additional constraints [Bandwidth (Multi) Coloring Problem, Bounded Vertex Coloring Problem] or an objective function with a special structure (Weighted Vertex Coloring Problem). The extension of the models for the classical VCP to the considered problems and the best performing algorithms from the literature, as well as the corresponding computational results, are reported.

Exact Solution of Graph Coloring Problems via Constraint Programming and Column Generation

“…gives the best results among those obtained with the exact methods, as reported in Malaguti and Toth (2010). Note that DSJC125.5, DSJC125.9, DSJC500.1c, flat300 20 0, and flat300 26 0 are solved to optimality for the first time with an exact method.…”

“…Moreover, we use the ILP solver also as a sort of primal heuristic: at each node of the search tree, we let CPLEX solve the restricted master problem with a branch-and-bound node limit set to 100 (see, parameter MIP node limit in CPLEX manuals), and with an upper bound set to the current upper bound -1. Table 6 shows the results we obtain for some hard instances that in the recent survey of Malaguti and Toth (2010) are reported as challenging for exact methods. The first six columns describe the instance features.…”

“…Table 3 reports the results of 11 instances that are solved within four minutes by the CP-LB approach, but that are not solved within two hours by Dsatur. Among these instances, ash958GPIA and wap05 are solved to optimality for the first time (with respect to the recent survey Malaguti and Toth (2010)), and apart from the first two instances, the others are considered as hard instances in Méndez-Díaz and Zabala (2008). For the Dsatur algorithm, the table gives the U B found at the time limit.…”

“…For three other instances, DSJC250.5, DSJC250.9, and DSJC500.9 we provide the best known lower bound obtained via column generation (see also Table 4) and the best upper bound obtained with an exact methods, though, for these last three instances, meta-heuristics provide better upper bounds. Malaguti and Toth (2010) and by heuristic algorithms (column"bks"). Note that for the instances queen10 10 and queen11 11 the optimum values are reported in Vasquez (2004).…”

“…Despite this interesting idea, the practical impact of this approach is limited. For a recent survey on exact and heuristic methods refer to Malaguti and Toth (2010).…”