Polyhedral studies of vertex coloring problems: The standard formulation

Despite the fact that many vertex coloring problems are polynomially solvable on certain graph classes, most of these problems are not "under control" from a polyhedral point of view. The equivalence between optimization and separation suggests the existence of integer programming formulat...

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Autores principales:	Delle Donne, Diego, Marenco, Javier Leonardo
Publicado:	2016
Materias:	Polyhedral characterization Standard formulation Vertex coloring Characterization Coloring Integer programming Problem solving Topology Block graphs Integer programming formulations Polyhedral studies Polynomially solvable Stable set polytope Valid inequality Vertex coloring problems Graph theory
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_15725286_v21_n_p1_DelleDonne http://hdl.handle.net/20.500.12110/paper_15725286_v21_n_p1_DelleDonne
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_15725286_v21_n_p1_DelleDonne
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spelling	paper:paper_15725286_v21_n_p1_DelleDonne2023-06-08T16:24:39Z Polyhedral studies of vertex coloring problems: The standard formulation Delle Donne, Diego Marenco, Javier Leonardo Polyhedral characterization Standard formulation Vertex coloring Characterization Coloring Integer programming Problem solving Topology Block graphs Integer programming formulations Polyhedral studies Polynomially solvable Stable set polytope Valid inequality Vertex coloring Vertex coloring problems Graph theory Despite the fact that many vertex coloring problems are polynomially solvable on certain graph classes, most of these problems are not "under control" from a polyhedral point of view. The equivalence between optimization and separation suggests the existence of integer programming formulations for these problems whose associated polytopes admit elegant characterizations. In this work we address this issue. As a starting point, we focus our attention on the well-known standard formulation for the classical vertex coloring problem. We present some general results about this formulation and we show that the vertex coloring polytope associated to this formulation for a graph G and a set of colors C corresponds to a face of the stable set polytope of a particular graph SG C. We further study the perfectness of SG C showing that when \|C\|>2, this graph is perfect if and only if G is a block graph, from which we deduce a complete characterization of the associated coloring polytopes for block graphs. We also derive a new family of valid inequalities generalizing several known families from the literature and we conjecture that this family is sufficient to completely describe the vertex coloring polytope associated to cacti graphs. © 2016 Elsevier B.V. All rights reserved. Fil:Delle Donne, D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Marenco, J. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2016 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_15725286_v21_n_p1_DelleDonne http://hdl.handle.net/20.500.12110/paper_15725286_v21_n_p1_DelleDonne
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Polyhedral characterization Standard formulation Vertex coloring Characterization Coloring Integer programming Problem solving Topology Block graphs Integer programming formulations Polyhedral studies Polynomially solvable Stable set polytope Valid inequality Vertex coloring Vertex coloring problems Graph theory
spellingShingle	Polyhedral characterization Standard formulation Vertex coloring Characterization Coloring Integer programming Problem solving Topology Block graphs Integer programming formulations Polyhedral studies Polynomially solvable Stable set polytope Valid inequality Vertex coloring Vertex coloring problems Graph theory Delle Donne, Diego Marenco, Javier Leonardo Polyhedral studies of vertex coloring problems: The standard formulation
topic_facet	Polyhedral characterization Standard formulation Vertex coloring Characterization Coloring Integer programming Problem solving Topology Block graphs Integer programming formulations Polyhedral studies Polynomially solvable Stable set polytope Valid inequality Vertex coloring Vertex coloring problems Graph theory
description	Despite the fact that many vertex coloring problems are polynomially solvable on certain graph classes, most of these problems are not "under control" from a polyhedral point of view. The equivalence between optimization and separation suggests the existence of integer programming formulations for these problems whose associated polytopes admit elegant characterizations. In this work we address this issue. As a starting point, we focus our attention on the well-known standard formulation for the classical vertex coloring problem. We present some general results about this formulation and we show that the vertex coloring polytope associated to this formulation for a graph G and a set of colors C corresponds to a face of the stable set polytope of a particular graph SG C. We further study the perfectness of SG C showing that when \|C\|>2, this graph is perfect if and only if G is a block graph, from which we deduce a complete characterization of the associated coloring polytopes for block graphs. We also derive a new family of valid inequalities generalizing several known families from the literature and we conjecture that this family is sufficient to completely describe the vertex coloring polytope associated to cacti graphs. © 2016 Elsevier B.V. All rights reserved.
author	Delle Donne, Diego Marenco, Javier Leonardo
author_facet	Delle Donne, Diego Marenco, Javier Leonardo
author_sort	Delle Donne, Diego
title	Polyhedral studies of vertex coloring problems: The standard formulation
title_short	Polyhedral studies of vertex coloring problems: The standard formulation
title_full	Polyhedral studies of vertex coloring problems: The standard formulation
title_fullStr	Polyhedral studies of vertex coloring problems: The standard formulation
title_full_unstemmed	Polyhedral studies of vertex coloring problems: The standard formulation
title_sort	polyhedral studies of vertex coloring problems: the standard formulation
publishDate	2016
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_15725286_v21_n_p1_DelleDonne http://hdl.handle.net/20.500.12110/paper_15725286_v21_n_p1_DelleDonne
work_keys_str_mv	AT delledonnediego polyhedralstudiesofvertexcoloringproblemsthestandardformulation AT marencojavierleonardo polyhedralstudiesofvertexcoloringproblemsthestandardformulation
_version_	1768542053595086848

Polyhedral studies of vertex coloring problems: The standard formulation

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