Note on the Polyhedral Description of the Minkowski Sum of Two L-convex Sets

TL;DR

This paper provides a detailed polyhedral description of Minkowski sums of L-convex sets, revealing their structure and connections to box-TDI polyhedra, with proofs and graph representations.

Contribution

It introduces the polyhedral description of L2-convex sets and shows their convex hulls are box-TDI polyhedra, supported by two distinct proofs.

Findings

Polyhedral description of L2-convex sets established

Convex hulls of L2-convex sets are box-TDI polyhedra

Graph representations of the results are provided

Abstract

L-convex sets are one of the most fundamental concepts in discrete convex analysis. Furthermore, the Minkowski sum of two L-convex sets, called L2-convex sets, is an intriguing object that is closely related to polymatroid intersection. This paper reveals the polyhedral description of an L2-convex set, together with the observation that the convex hull of an L2-convex set is a box-TDI polyhedron. Two different proofs are given for the polyhedral description. The first is a structural short proof, relying on the conjugacy theorem in discrete convex analysis, and the second is a direct algebraic proof, based on Fourier-Motzkin elimination. The obtained results admit natural graph representations. Implications of the obtained results in discrete convex analysis are also discussed.