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{0, 1/2}-Chvátal-Gomory cuts

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Abstract

Given the integer polyhedronP t := conv{x ∈ℤn:Ax⩽b}, whereA ∈ℤm × n andb ∈ℤm, aChvátal-Gomory (CG)cut is a valid inequality forP 1 of the type λτAx⩽⌊λτb⌋ for some λ∈ℝ m+ such that λτA∈ℤn . In this paper we study {0, 1/2}-CG cuts, arising for λ∈{0, 1/2}m. We show that the associated separation problem, {0, 1/2}-SEP, is equivalent to finding a minimum-weight member of a binary clutter. This implies that {0, 1/2}-SEP is NP-complete in the general case, but polynomially solvable whenA is related to the edge-path incidence matrix of a tree. We show that {0, 1/2}-SEP can be solved in polynomial time for a convenient relaxation of the systemAx<-b. This leads to an efficient separation algorithm for a subclass of {0, 1/2}-CG cuts, which often contains wide families of strong inequalities forP 1. Applications to the clique partitioning, asymmetric traveling salesman, plant location, acyclic subgraph and linear ordering polytopes are briefly discussed.

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Author notes

  1. Matteo Fischetti

    Present address: Dipartimento di Elettronica e Informatica, Università Degli Studi di Padova, Via Gradenigo 6/A, 35100, Padova, Italy

Authors and Affiliations

  1. DEIS, University of Bologna, Italy

    Alberto Caprara

  2. DMI, University of Udine, Italy

    Matteo Fischetti

Authors
  1. Alberto Caprara
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  2. Matteo Fischetti
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Caprara, A., Fischetti, M. {0, 1/2}-Chvátal-Gomory cuts. Mathematical Programming 74, 221–235 (1996). https://doi.org/10.1007/BF02592196

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  • DOI: https://doi.org/10.1007/BF02592196

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