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The multi-terminal vertex separator problem: Polyhedral analysis and Branch-and-Cut

Cornaz, Denis; Magnouche, Youcef; Mahjoub, Ali Ridha; Martin, Sébastien (2019), The multi-terminal vertex separator problem: Polyhedral analysis and Branch-and-Cut, Discrete Applied Mathematics, 256, 1, p. 11-37. 10.1016/j.dam.2018.10.005

Type
Article accepté pour publication ou publié
Date
2019
Journal name
Discrete Applied Mathematics
Volume
256
Number
1
Publisher
Elsevier
Pages
11-37
Publication identifier
10.1016/j.dam.2018.10.005
Metadata
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Author(s)
Cornaz, Denis
Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Magnouche, Youcef
Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Mahjoub, Ali Ridha
Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Martin, Sébastien cc
Laboratoire de Conception, Optimisation et Modélisation des Systèmes [LCOMS]
Abstract (EN)
In this paper we consider a variant of the k-separator problem. Given a graph G=(V∪T,E) with V∪T the set of vertices, where T is a set of k terminals, the multi-terminal vertex separator problem consists in partitioning V∪T into k+1 subsets {S,V1,...,Vk} such that there is no edge between two different subsets Vi and Vj, each Vi contains exactly one terminal and the size of S is minimum. In this paper, we first show that the problem is NP-hard. Then we give two integer programming formulations for the problem. For one of these formulations, we investigate the related polyhedron and discuss its polyhedral structure. We describe some valid inequalities and characterize when these inequalities define facets. We also derive separation algorithms for these inequalities. Using these results, we develop a Branch-and-Cut algorithm for the problem, along with an extensive computational study.
Subjects / Keywords
Vertex separator problem; Integer programming; Polytope; Facet; Branch-and-Cut algorithm; Complexity; Separation algorithm

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