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We consider the problem of orthogonally packing a given set of rectangular items into a given strip, by minimizing the overall height of the packing. The problem is NP-hard in the strong sense, and finds several applications in cutting and packing. We propose a new relaxation that produces good lower bounds and gives information to obtain effective heuristic algorithms. These results are used in a branch-and-bound algorithm, which was able to solve test instances from the literature involving up to 200 items.

a b s t r a c tGiven an undirected graph G = (V , E), the Vertex Coloring Problem (VCP) requires to assign a color to each vertex in such a way that colors on adjacent vertices are different and the number of colors used is minimized. In this paper, we present an exact algorithm for the solution of VCP based on the well-known Set Covering formulation of the problem. We propose a Branch-and-Price algorithm embedding an effective heuristic from the literature and some methods for the solution of the slave problem, as well as two alternative branching schemes. Computational experiments on instances from the literature show the effectiveness of the algorithm, which is able to solve, for the first time to proven optimality, five of the benchmark instances in the literature, and reduce the optimality gap of many others.

Given an undirected graph G = V E, the vertex coloring problem (VCP) requires to assign a color to each vertex in such a way that colors on adjacent vertices are different and the number of colors used is mini- mized. In this paper, we propose a metaheuristic approach for VCP that performs two phases: the first phase is based on an evolutionary algorithm, whereas the second one is a postoptimization phase based on the set covering formulation of the problem. Computational results on a set of DIMACS instances show that the over- all algorithm is able to produce high-quality solutions in a reasonable amount of time. For four instances, the proposed algorithm is able to improve the best-known solution while for almost all the remaining instances, it finds the best-known solution in the literature

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