Some Valid Inequalities for the Set Partitioning Problem

Abstract

We introduce a family of inequalities derived from the logical implications of set partitioning constraints and investigate their properties and potential uses. We start with a class of homogeneous canonical inequalities that we call elementary, and discuss conditions under which they are (a) valid, (b) cutting planes, (c) maximal, and (d) facets or improper faces of the set partitioning polytope. We give two procedures for strengthening nonmaximal valid elementary inequalities. Next we derive two nonhomogeneous equivalents of the elementary inequalities, which are of the set packing and set covering types respectively. Using the first of these equivalents, we introduce a “strong” intersection graph, a supergraph of the (common) intersection graph, whose facet generating subgraphs (cliques, odd holes, etc.) give rise to valid inequalities for the set partitioning problem. These inequalities subsume or dominate the similar inequalities that one can derive for the associated set packing problem. One subclass can be used to enhance orthogonality tests in implicit enumeration or column generating algorithms. Further, we introduce two types of composite inequalities, obtainable by combining elementary inequalities according to specific rules, and some related inequalities obtainable directly from the set partitioning constraints. These inequalities provide convenient primal all-integer cutting planes that offer a greater flexibility and are usually stronger than the earlier cuts which do not use the special structure of the set partitioning problem. In the final section we discuss a primal algorithm which uses these cuts in conjunction with implicit enumeration.