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

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Integer Programming and Combinatorial Optimization (IPCO 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,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.

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Author information

Authors and Affiliations

  1. IMAS-CONICET and Departamento de Computación, FCEN, Universidad de Buenos Aires, Argentina

    Flavia Bonomo

  2. Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università Tor Vergata, Roma, Italy

    Gianpaolo Oriolo & Claudia Snels

  3. Bordeaux Institute of Mathematics, France

    Gautier Stauffer

Authors
  1. Flavia Bonomo
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  2. Gianpaolo Oriolo
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  3. Claudia Snels
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  4. Gautier Stauffer
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Editor information

Editors and Affiliations

  1. Massachusetts Institute of Technology, 77 Massachusetts Ave., 02139, Cambridge, MA, USA

    Michel Goemans

  2. Universidad de Chile, Santiago, Chile

    José Correa

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Bonomo, F., Oriolo, G., Snels, C., Stauffer, G. (2013). Minimum Clique Cover in Claw-Free Perfect Graphs and the Weak Edmonds-Johnson Property. In: Goemans, M., Correa, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2013. Lecture Notes in Computer Science, vol 7801. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36694-9_8

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  • DOI: https://doi.org/10.1007/978-3-642-36694-9_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-36693-2

  • Online ISBN: 978-3-642-36694-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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