Network Formulations of Mixed-Integer Programs

We consider mixed-integer sets of the type MIX = {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 setMIX is NP-complete even in the case in which the system describes mixed-integer network flows with half-integral requirements on the nodes. This is in contrast to the case where A is totally unimodular and contains at most two nonzeros per row. Denoting such mixed-integer sets by MIX , we provide an extended formulation for the convex hull of MIX whose constraint matrix is a dual network matrix with an integral right-hand-side vector. The size of this formulation depends on the number of distinct fractional parts taken by the continuous variables in the extreme points of conv(MIX ). When this number is polynomial in the dimension of the matrix A, the extended formulation is of polynomial size. If, in addition, the corresponding list of fractional parts can be computed efficiently, then our result provides a polynomial algorithm for the optimization problem over MIX . We show that there are instances for which this list is of exponential size, and we also give conditions under which it is short and can be efficiently computed. Finally we show that these results for the set MIX 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.