Network Formulations of Mixed-Integer Programs

CORE Discussion Paper 2006/117

20 Pages Posted: 14 Mar 2007

See all articles by Michele Conforti

Michele Conforti

University of Padua - Department of Pure and Applied Mathematics

Marco Di Summa

University of Padua - Department of Pure and Applied Mathematics

Fritz Eisenbrand

University of Paderborn - Faculty of Computer Science, Electrical Engineering and Mathematics

Laurence A. Wolsey

Center for Operations Research and Econometrics (CORE)

Date Written: December 2006

Abstract

We consider mixed-integer sets of the type MIX TU = {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 MIX TU 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 MIX 2TU, we provide an extended formulation for the convex hull of MIX 2TU 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(MIX 2TU). When this number is polynomial in the dimension of the matrix A, the formulation is of polynomial size and the optimization problem over MIX 2TU 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 MIX 2TU 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.

Keywords: mixed-integer set, totally unimodular matrix, extended formulation, convex hull, dual of network matrix

Suggested Citation

Conforti, Michele and Di Summa, Marco and Eisenbrand, Fritz and Wolsey, Laurence A., Network Formulations of Mixed-Integer Programs (December 2006). CORE Discussion Paper 2006/117, Available at SSRN: https://ssrn.com/abstract=970911 or http://dx.doi.org/10.2139/ssrn.970911

Michele Conforti (Contact Author)

University of Padua - Department of Pure and Applied Mathematics ( email )

Padova, 35100
Italy

Marco Di Summa

University of Padua - Department of Pure and Applied Mathematics ( email )

Padova, 35100
Italy

Fritz Eisenbrand

University of Paderborn - Faculty of Computer Science, Electrical Engineering and Mathematics ( email )

Warburger Str. 100
D-33098 Paderborn
Germany

Laurence A. Wolsey

Center for Operations Research and Econometrics (CORE) ( email )

34 Voie du Roman Pays
1348 Louvain-la-Neuve, 1348
Belgium