A Discrete Convex Min-Max Formula for Box-TDI Polyhedra

TL;DR

This paper establishes a min-max formula for integer-valued discrete convex functions over box-TDI polyhedra, unifying various combinatorial optimization problems and introducing new theoretical insights.

Contribution

It introduces a novel min-max theorem for discrete convex functions on box-TDI polyhedra, including variants with and without conjugates, and applies to diverse combinatorial optimization problems.

Findings

Proves a min-max formula for discrete convex minimization over box-TDI polyhedra.

Unifies several classes of combinatorial optimization problems under a common framework.

Shows applicability to inverse combinatorial optimization problems.

Abstract

A min-max formula is proved for the minimum of an integer-valued separable discrete convex function where the minimum is taken over the set of integral elements of a box total dual integral (box-TDI) polyhedron. One variant of the theorem uses the notion of conjugate function (a fundamental concept in non-linear optimization) but we also provide another version that avoids conjugates, and its spirit is conceptually closer to the standard form of classic min-max theorems in combinatorial optimization. The presented framework provides a unified background for separable convex minimization over the set of integral elements of the intersection of two integral base-polyhedra, submodular flows, L-convex sets, and polyhedra defined by totally unimodular (TU) matrices. As an unexpected application, we show how a wide class of inverse combinatorial optimization problems can be covered by this new framework.