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Network formulations of mixed-integer programs

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  • CONFORTI, Michele
  • DI SUMMA, Marco
  • EISENBRAND, Fritz
  • WOLSEY, Laurence A.

Abstract

We consider mixed-integer sets of the type M IX T U = {x : Ax b; xi integer, i I}, where A is a totally unimodular matrix, b is an arbitrary vector and I is a nonempty subset of the column indices of A. We show that the problem of checking nonemptiness of a set M IX T U is NP-complete when A contains at most two nonzeros per column. This is in contrast to the case when A is TU and contains at most two nonzeros per row. Denoting the set by M IX 2T U , we provide an extended formulation for the convex hull of M IX 2T U whose constraint matrix is the dual of a network matrix, and with integer right hand side vector. The size of this formulation depends on the number |F | of distinct fractional parts taken by the continuous variables in the extreme points of conv(M IX 2T U ). When this number is polynomial in the dimension of the matrix A, the formulation is of polynomial size and the optimization problem over M IX 2T U lies in P. We show that there are instances for which |F | is of exponential size, and we also give conditions under which |F | is of polynomial size. Finally we show that these results for the set M IX 2T U provide a unified framework leading to polynomial-size extended formulations for several generalizations of mixing sets and lot-sizing sets studied in the last few years.

Suggested Citation

  • CONFORTI, Michele & DI SUMMA, Marco & EISENBRAND, Fritz & WOLSEY, Laurence A., 2006. "Network formulations of mixed-integer programs," LIDAM Discussion Papers CORE 2006117, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  • Handle: RePEc:cor:louvco:2006117
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    References listed on IDEAS

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    1. GÜNLÜK, Oktay & POCHET, Yves, 2001. "Mixing mixed-integer inequalities," LIDAM Reprints CORE 1504, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. VAN VYVE, Mathieu, 2006. "Linear-programming extended formulations for the single-item lot-sizing problem with backlogging and constant capacity," LIDAM Reprints CORE 1855, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Pochet, Y. & Wolsey, L. A., 1994. "Polyhedra for lot-sizing with Wagner-Whitin costs," LIDAM Reprints CORE 1129, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Mathieu Van Vyve, 2005. "The Continuous Mixing Polyhedron," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 441-452, May.
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