Minimum Clique Cover in Claw-Free Perfect Graphs and the Weak Edmonds-Johnson Property

  • Flavia Bonomo
  • Gianpaolo Oriolo
  • Claudia Snels
  • Gautier Stauffer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7801)

Abstract

We give new algorithms for the minimum (weighted) clique cover in a claw-free perfect graph G, improving the complexity from O(|V(G)|5) to O(|V(G)|3). The new algorithms build upon neat reformulations of the problem: it basically reduces either to solving a 2-SAT instance (in the unweighted case) or to testing if a polyhedra associated with the edge-vertex incidence matrix of a bidirected graph has an integer solution (in the weighted case). The latter question was elegantly answered using neat polyhedral arguments by Schrijver in 1994. We give an alternative approach to this question combining pure combinatorial arguments (using techniques from 2-SAT and shortest paths) with polyhedral ones. Our approach is inspired by an algorithm from the Constraint Logic Programming community and we give as a side benefit a formal proof that the corresponding algorithm is correct (apparently answering an open question in this community). Interestingly, the systems we study have properties closely connected with the so-called Edmonds-Johnson property and we study some interesting related questions.

Keywords

clique cover claw-free perfect graphs bidirected graphs Edmonds-Johnson property 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Flavia Bonomo
    • 1
  • Gianpaolo Oriolo
    • 2
  • Claudia Snels
    • 2
  • Gautier Stauffer
    • 3
  1. 1.IMAS-CONICET and Departamento de Computación, FCENUniversidad de Buenos AiresArgentina
  2. 2.Dipartimento di Ingegneria Civile e Ingegneria InformaticaUniversità Tor VergataRomaItaly
  3. 3.Bordeaux Institute of MathematicsFrance

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