Mathematical Programming

, Volume 44, Issue 1, pp 181–202

On the facial structure of the set covering polytope

  • A. Sassano
Article

DOI: 10.1007/BF01587087

Cite this article as:
Sassano, A. Mathematical Programming (1989) 44: 181. doi:10.1007/BF01587087

Abstract

Given a bipartite graphG = (V, U, E), a cover ofG is a subset\(D \subseteq V\) with the property that each nodeu ∈ U is adjacent to at least one nodevD. If a positive weightcv is associated with each nodevV, the covering problem (CP) is to find a cover ofG having minimum total weight.

In this paper we study the properties of the polytopeQ(G) ⊂ ℝ|V|, the convex hull of the incidence vectors of all the covers inG. After discussing some general properties ofQ(G) we introduce a large class of bipartite graphs with special structure and describe several types of rank facets of the associated polytopes.

Furthermore we present two lifting procedures to derive valid inequalities and facets of the polytopeQ(G) from the facets of any polytopeQ(G′) associated with a subgraphG′ ofG. An example of the application of the theory to a class of hard instances of the covering problem is also presented.

Copyright information

© The Mathematical Programming Society, Inc. 1989

Authors and Affiliations

  • A. Sassano
    • 1
  1. 1.Istituto di Analisi dei Sistemi ed Informatica del CNRRomaItaly

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