{"title":"ILP Models and Column Generation for the Minimum Sum Coloring Problem","authors":"Fabio Furini, Enrico Malaguti , Sébastien Martin, Ian-Christopher Ternier","doi":"10.1016/j.endm.2018.01.023","DOIUrl":null,"url":null,"abstract":"<div><p>We study two Integer Linear Programming (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. The first ILP formulation has a polynomial number of variables while the second one has an exponential number of variables and is tackled via column generation. Computational tests show that the linear programming relaxation of the second formulation provides tight lower bounds which allow us to solve to proven optimality some hard instances of the literature.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":"64 ","pages":"Pages 215-224"},"PeriodicalIF":0.0000,"publicationDate":"2018-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.01.023","citationCount":"7","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Notes in Discrete Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1571065318300234","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 7
Abstract
We study two Integer Linear Programming (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. The first ILP formulation has a polynomial number of variables while the second one has an exponential number of variables and is tackled via column generation. Computational tests show that the linear programming relaxation of the second formulation provides tight lower bounds which allow us to solve to proven optimality some hard instances of the literature.
期刊介绍：
Electronic Notes in Discrete Mathematics is a venue for the rapid electronic publication of the proceedings of conferences, of lecture notes, monographs and other similar material for which quick publication is appropriate. Organizers of conferences whose proceedings appear in Electronic Notes in Discrete Mathematics, and authors of other material appearing as a volume in the series are allowed to make hard copies of the relevant volume for limited distribution. For example, conference proceedings may be distributed to participants at the meeting, and lecture notes can be distributed to those taking a course based on the material in the volume.
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