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Mathematical Programming

, Volume 36, Issue 2, pp 157–173 | Cite as

On the cut polytope

  • Francisco Barahona
  • Ali Ridha Mahjoub
Article

Abstract

The cut polytopeP C (G) of a graphG=(V, E) is the convex hull of the incidence vectors of all edge sets of cuts ofG. We show some classes of facet-defining inequalities ofP C (G). We describe three methods with which new facet-defining inequalities ofP C (G) can be constructed from known ones. In particular, we show that inequalities associated with chordless cycles define facets of this polytope; moreover, for these inequalities a polynomial algorithm to solve the separation problem is presented. We characterize the facet defining inequalities ofP C (G) ifG is not contractible toK 5. We give a simple characterization of adjacency inP C (G) and prove that for complete graphs this polytope has diameter one and thatP C (G) has the Hirsch property. A relationship betweenP C (G) and the convex hull of incidence vectors of balancing edge sets of a signed graph is studied.

Key words

Max cut problem facets of polyhedra polyhedral combinatorics 

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Copyright information

© The Mathematical Programming Society, Inc. 1986

Authors and Affiliations

  • Francisco Barahona
    • 1
  • Ali Ridha Mahjoub
    • 2
  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  2. 2.Laboratoire ARTEMIS, Institut IMAGUniversité Scientifique et Medicale de GrenobleFrance

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