Polyhedral results for two classes of submodular sets with GUB constraints

TL;DR

This paper develops strong, facet-defining inequalities for two classes of submodular sets with GUB constraints, improving convex hull descriptions and computational efficiency for related optimization problems.

Contribution

It introduces a new class of lifted inequalities stronger than EPIs, with linear-time computability and complete convex hull characterizations for the sets.

Findings

Lifted inequalities are facet-defining and stronger than EPIs.

Inequalities can be computed in linear time.

Computational results show improved performance in optimization problems.

Abstract

In this paper, we investigate the polyhedral structure of two submodular sets with generalized upper bound (GUB) constraints, which arise as important substructures in various real-world applications. We derive a class of strong valid inequalities for the two sets using sequential lifting techniques. The proposed lifted inequalities are facet-defining for the convex hulls of two sets and are stronger than the well-known extended polymatroid inequalities (EPIs). We provide a more compact characterization of these inequalities and show that each of them can be computed in linear time. Moreover, the proposed lifted inequalities, together with bound and GUB constraints, can completely characterize the convex hulls of the two sets, and can be separated using a combinatorial polynomial-time algorithm. Finally, computational results on probabilistic covering location and multiple probabilistic knapsack problems demonstrate the superiority of the proposed lifted inequalities over the EPIs within a branch-and-cut framework.