Cutting planes from conditional bounds: A new approach to set covering

Abstract

A conditional lower bound on the minimand of an integer program is a number which would be a valid lower bound if the constraint set were amended by certain inequalities, also called conditional. If such a conditional lower bound exceeds some known upper bound, then every solution better than the one corresponding to the upper bound violates at least one of the conditional inequalities. This yields a valid disjunction, which can be used to partition the feasible set, or to derive a family of valid cutting planes. In the case of a set covering problem, these cutting planes are themselves of the set covering type. The family of valid inequalities derived from conditional bounds subsumes as a special case the Bellmore-Ratliff inequalities generated via involutory bases, but is richer than the latter class and contains considerably stronger members, where strength is measured by the number of positive coefficients. We discuss the properties of the family of cuts from conditional bounds, and give a procedure for generating strong members of the family. Finally, we outline a class of algorithms based on these cuts. Our approach was implemented and extensively tested in a computational study whose results are reported in a companion paper [2]. The algorithm that emerged from the testing seems capable of solving considerably larger set covering problem than earlier methods.

Research supported by the National Science Foundation under grant MCS 76-12026 A02 and the Office of Naval Research under contract N00014-75-C-0621 NR 047-048.