research-article

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

Authors:

S. MartinAuthors Info & Claims

Volume 256, Issue C

Pages 11 - 37

https://doi.org/10.1016/j.dam.2018.10.005

Published: 15 March 2019 Publication History

Abstract

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, V 1, …, V k } such that there is no edge between two different subsets V i and V j, each V i 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.

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Digital Library

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https://lemon.cs.elte.hu/trac/lemon.

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http://www.boost.org/doc/libs/1_60_0/libs/graph/.

Cited By

Irmai JZhao SSchöne MPresberger JAndres B(2024)A Graph Multi-separator Problem for Image SegmentationJournal of Mathematical Imaging and Vision10.1007/s10851-024-01201-166:5(839-872)Online publication date: 1-Oct-2024
https://dl.acm.org/doi/10.1007/s10851-024-01201-1
Hu HLi XWu J(2023)A note on the SDP relaxation of the minimum cut problemJournal of Global Optimization10.1007/s10898-022-01235-y87:2-4(857-876)Online publication date: 1-Nov-2023
https://dl.acm.org/doi/10.1007/s10898-022-01235-y

Index Terms

The multi-terminal vertex separator problem: Polyhedral analysis and Branch-and-Cut
1. Mathematics of computing
  1. Discrete mathematics
  2. Mathematical analysis
    1. Mathematical optimization
      1. Mixed discrete-continuous optimization
        Integer programming
2. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization
      1. Mixed discrete-continuous optimization
        Integer programming

Index terms have been assigned to the content through auto-classification.

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Published In

cover image Discrete Applied Mathematics

Discrete Applied Mathematics Volume 256, Issue C

Mar 2019

163 pages

ISSN:0166-218X

Issue’s Table of Contents

Elsevier B.V.

Publisher

Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 15 March 2019

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Cited By

Irmai JZhao SSchöne MPresberger JAndres B(2024)A Graph Multi-separator Problem for Image SegmentationJournal of Mathematical Imaging and Vision10.1007/s10851-024-01201-166:5(839-872)Online publication date: 1-Oct-2024
https://dl.acm.org/doi/10.1007/s10851-024-01201-1
Hu HLi XWu J(2023)A note on the SDP relaxation of the minimum cut problemJournal of Global Optimization10.1007/s10898-022-01235-y87:2-4(857-876)Online publication date: 1-Nov-2023
https://dl.acm.org/doi/10.1007/s10898-022-01235-y

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