A polyhedral study of the maximum stable set problem with weights on vertex-subsets

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• 作者：Manoel Campê ; lo^a ; ^{mcampelo@lia.ufc.br" class="auth_mail" title="E-mail the corresponding author} ; Victor A. Campos^b ; ^{campos@lia.ufc.br" class="auth_mail" title="E-mail the corresponding author} ; Ricardo C. Corrê ; a^b ; ^{correa@lia.ufc.br" class="auth_mail" title="E-mail the corresponding author} ; Diego Delle Donne^c ; ^{ddelledo@ungs.edu.ar" class="auth_mail" title="E-mail the corresponding author} ; Javier Marenco^c ; ^{jmarenco@ungs.edu.ar" class="auth_mail" title="E-mail the corresponding author} ; Marcelo Mydlarz^c ; ^{mmydlarz@ungs.edu.ar" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Maximum stable set ; Integer programming
• 刊名：Discrete Applied Mathematics
• 出版年：2016
• 出版时间：10 September 2016
• 年：2016
• 卷：210
• 期：Complete
• 页码：223-234
• 全文大小：509 K

文摘

Given a graph class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15002826&_mathId=si20.gif&_user=111111111&_pii=S0166218X15002826&_rdoc=1&_issn=0166218X&md5=b05dea4bdbf74ffdf9dde0fe8898158d" title="Click to view the MathML source">G=(V,E)class="mathContainer hidden">class="mathCode">$G=(V,E)$, a family of nonempty vertex-subsets class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15002826&_mathId=si21.gif&_user=111111111&_pii=S0166218X15002826&_rdoc=1&_issn=0166218X&md5=9356e5797b85687890e7b5d62d2c3c11" title="Click to view the MathML source">S⊆2^Vclass="mathContainer hidden">class="mathCode">$\mathcal{S}\subseteq 2^V$, and a weight class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15002826&_mathId=si22.gif&_user=111111111&_pii=S0166218X15002826&_rdoc=1&_issn=0166218X&md5=755b178ecbb3fd6f82af105611a21148" title="Click to view the MathML source">w:S→R₊class="mathContainer hidden">class="mathCode">$w:\mathcal{S}\to {\mathbb{R}}_{+}$, the maximum stable set problem with weights on vertex-subsets   consists in finding a stable set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15002826&_mathId=si23.gif&_user=111111111&_pii=S0166218X15002826&_rdoc=1&_issn=0166218X&md5=cf531214151137c1f627179a9891930a" title="Click to view the MathML source">Iclass="mathContainer hidden">class="mathCode">$I$ of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15002826&_mathId=si24.gif&_user=111111111&_pii=S0166218X15002826&_rdoc=1&_issn=0166218X&md5=6bfa4ecccecbd5aa186b9accee5e46f5" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode">$G$ maximizing the sum of the weights of the sets in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15002826&_mathId=si10.gif&_user=111111111&_pii=S0166218X15002826&_rdoc=1&_issn=0166218X&md5=da977c5fdddd9eb3cc9af52d45df5234" title="Click to view the MathML source">Sclass="mathContainer hidden">class="mathCode">$\mathcal{S}$ that intersect class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15002826&_mathId=si23.gif&_user=111111111&_pii=S0166218X15002826&_rdoc=1&_issn=0166218X&md5=cf531214151137c1f627179a9891930a" title="Click to view the MathML source">Iclass="mathContainer hidden">class="mathCode">$I$. This problem arises within a natural column generation approach for the vertex coloring problem. In this work we perform an initial polyhedral study of this problem, by introducing a natural integer programming formulation and studying the associated polytope. We address general facts on this polytope including some lifting results, we provide connections with the stable set polytope, and we present three families of facet-inducing inequalities.