Graph coloring is the problem of assigning a minimum number of colors to all vertices of a graph such that no two adjacent vertices receive the same color. The selective graph coloring problem is a generalization of the standard graph coloring problem; given a graph with a partition of its vertex set into clusters, the objective is to choose exactly one vertex per cluster so that, among all possible selections, the number of colors necessary to color the vertices in the selection is minimum. This study focuses on a decomposition based exact solution framework for selective coloring in some perfect graph families: in particular, permutation, generalized split, and chordal graphs where the selective coloring problem is known to be NP‐hard. Our method combines integer programming techniques and combinatorial algorithms for the graph classes of interest. We test our method on graphs with different sizes and densities, present computational results and compare them with solving an integer programming formulation of the problem by CPLEX, and a state‐of‐the art algorithm from the literature. Our computational experiments indicate that our decomposition approach significantly improves solution performance in low‐density graphs, and regardless of edge‐density in the class of chordal graphs.

We investigate the maximum induced matching problem (MIM), which is the problem of finding an induced matching having the largest cardinality on an undirected graph. The problem is known to be NP-hard for general graphs. We first propose a vertex-based integer programming formulation for MIM, which is more compact compared to an edge-based formulation found in the literature. We also introduce the maximum weight induced matching problem (MWIM), which generalizes MIM so that vertices and edges have weights. We adapt the edge-based formulation to MWIM, and propose a quadratic programming formulation of MWIM based on our vertex-based formulation. We then linearize our quadratic programming formulation, and devise a Benders decomposition algorithm that exploits a special structure of the linearized formulation. We also propose valid inequalities and formulation tightening procedures to improve the efficiency of our approach. Our computational tests on a large suite of randomly generated graphs show that our vertex-based formulation and decomposition approach significantly improve the solvability of MIM and MWIM, especially on dense graphs. The online appendix and data are available at https://doi.org/10.1287/ijoc.2017.0764 .

Abstract– We investigate the problem of finding a maximal matching that has minimum total weight on a given edge‐weighted graph. Although the minimum weight maximal matching problem is NP‐hard in general, polynomial time exact or approximation algorithms on several restricted graph classes are given in the literature. In this article, we propose an exact algorithm for solving several variants of the problem on general graphs. In particular, we develop integer programming (IP) formulations for the problem and devise a decomposition algorithm, which is based on a combination of IP techniques and combinatorial matching algorithms. Our computational tests on a large suite of randomly generated graphs show that our decomposition approach significantly improves the solvability of the problem compared to the underlying IP formulation. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol. 62(4), 273–287 2013