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Exact algorithms for the Vertex Coloring Problem and its generalisations

Abstract : Given an undirected graph, the Vertex Coloring Problem (VCP) consists of assigning a color to each vertex of the graph such that two adjacent vertices do not share the same color and the total number of colors is minimized. DSATUR is an effective exact algorithm for the VCP. We introduce new lower bounding techniques enabling the computing of a lower bound at each node of the branching scheme. Our new DSATUR outperforms the state of the art for random VCP instances with high density, significantly increasing the size of solvable instances. Similar results can be achieved for a subset of high density DIMACS instances. We study three ILP formulations for the Minimum Sum Coloring Problem (MSCP). The problem is an extension of the classical Vertex Coloring Problem in which each color is represented by a positive natural number. The MSCP asks to minimize the sum of the cardinality of subsets of vertices receiving the same color, weighted by the index of the color, while ensuring that vertices linked by an edge receive different colors. We focus on studying an extended formulation and devise a complete Branch-and-Price algorithm.
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Ian-Christopher Ternier. Exact algorithms for the Vertex Coloring Problem and its generalisations. Operations Research [cs.RO]. Université Paris sciences et lettres, 2017. English. ⟨NNT : 2017PSLED060⟩. ⟨tel-01867961⟩

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