On the Separation of Maximally Violated mod-k Cuts

Abstract

Separation is of fundamental importance in cutting-plane based techniques for Integer Linear Programming (ILP). In recent decades, a considerable research effort has been devoted to the definition of effective separation procedures for families of well-structured cuts. In this paper we address the separation of Chvátal rank-1 inequalities in the context of general ILP’s of the form min{c ^T x : Ax ≤ b, x integer}, where A is an m × n integer matrix and b an m-dimensional integer vector. In particular, for any given integer k we study mod-k cuts of the form γ ^T Ax ≤ ⌊γ ^T b⌋ for any γ ∈ {0, 1/k, . . ., (k − 1)/k}^m such that γ ^T is integer. Following the line of research recently proposed for mod- 2 cuts by Applegate, Bixby, Chvátal and Cook [1] and Fleischer and Tardos [16], we restrict to maximally violated cuts, i.e., to inequalities which are violated by (k — 1)/k by the given fractional point. We show that, for any given k, such a separation requires O(mnmin{m, n}) time. Applications to the TSP are discussed. In particular, for any given k, we propose an O(|V|²|E*|)-time exact separation algorithm for mod-k cuts which are maximally violated by a given fractional TSP solution with support graph G* = (V,E*). This implies that we can identify a maximally violated TSP cut whenever a maximally violated (extended). comb inequality exists. Finally, specific classes of (sometimes new) facet-defining mod-k cuts for the TSP are analyzed.