On the separation of maximally violated mod-k cuts

Caprara, A and Fischetti, M and Letchford, A N (1999) On the separation of maximally violated mod-k cuts. In: Integer Programming and Combinatorial Optimization : Proceedings of the 7th International IPCO Conference. Lecture Notes in Computer Science . Springer, AUT, pp. 87-98. ISBN 978-3-540-66019-4

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 Chvatal rank-1 inequalities in the context of general ILP's of the form min c^Tx : Ax <= b; x integer, where A is an m x 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 (lambda^TA)x <= floor lambda^Tb floor¸ for any lambda in {0,1/2}^m such that lambda^TA is integer. Following the line of research recently proposed for mod-2 cuts by Applegate, Bixby, Chvatal and Cook, and Fleischer and Tardos, 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(mn min{m,n}) time. Applications to the TSP are discussed.