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Facet Generating Techniques

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Research Trends in Combinatorial Optimization

Summary

A major goal of polyhedral combinatorics is to find classes of essential, i.e. facet inducing, inequalities which describe a combinatorially defined polyhedron P. In general this is a difficult task. We consider the case in which we have knowledge of facets for a face F of P, and present a general technique for exploiting the close relationship between such polyhedra in order to obtain facets for P from facets of F. We demonstrate the usefulness of this technique by applying it in three instances where this relationship holds, namely the linear ordering polytope and the acyclic subgraph polytope, the asymmetric travelling salesman polytope and the monotone asymmetric travelling salesman polytope, and the symmetric travelling salesman polytope and the two-connected spanning subgraph polytope. In the last case we obtain a class of facet inducing inequalities for the two-connected spanning subgraph polytope by our procedure. This technique has also been applied by Boyd and Hao (1993) to show a class of inequalities is facet inducing for the two edge connected spanning subgraph polytope, by Leung and Lee (1994) to show a class of inequalities is facet inducing for the acyclic subgraph polytope and by Bauer (1997) to show several classes of inequalities are facet inducing for the circuit polytope.

The above technique requires that we demonstrate validity of an inequality. We discuss the problem of proving an inequality is valid for the integer hull of a polyhedron, and show that this problem is in NP for classes of polyhedra having bounded Chvátal–Gomory rank. This has the following consequence. Suppose we have an integer programming formulation of the members of an NP-complete class of problems with the property that we can, in polynomial time, show the validity of our defining inequalities. Then there will be problems in the class for which a linear system sufficient to define the integer hull will necessarily contain an inequality of arbitrarily large Chvátal–Gomory rank unless NP = co-NP.

Research partially supported by grants from N.S.E.R.C. of Canada.

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Authors and Affiliations

  1. School of Information Technology and Engineering, University of Ottawa, 800 King Edward Ave., Ottawa, Ontario, K1N 6N5, Canada

    Sylvia Boyd

  2. Center for Business Optimization, IBM Global Business Services, Route 100, Somers, NY, 10589, USA

    William R. Pulleyblank

Authors
  1. Sylvia Boyd
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  2. William R. Pulleyblank
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Editors and Affiliations

  1. Industrial and Systems Engineering, Georgia Tech, 765 Ferst Drive, Atlanta, Georgia, 30332-0205, USA

    William Cook

  2. Institute of Mathematics, Eötvös Loránd University, Pázmány Péter sétány 1/C, H-1117, Budapest, Hungary

    László Lovász

  3. Research Institute for Discrete Mathematics, University of Bonn, Lennéstr. 2, 53113, Bonn, Germany

    Jens Vygen

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Boyd, S., Pulleyblank, W.R. (2009). Facet Generating Techniques. In: Cook, W., Lovász, L., Vygen, J. (eds) Research Trends in Combinatorial Optimization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76796-1_2

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