On Equivalence of M$^\natural$-concavity of a Set Function and Submodularity of Its Conjugate

TL;DR

This paper provides a new proof of the fundamental equivalence between M$^ atural$-concavity of set functions and submodularity of their conjugates, a key result in discrete convex analysis.

Contribution

The paper introduces a novel proof of the equivalence between M$^ atural$-concavity and submodularity, enhancing theoretical understanding in discrete convex analysis.

Findings

New proof of the equivalence between M$^ atural$-concavity and submodularity

Reinforces the theoretical foundation of discrete convex analysis

Clarifies the relationship between set functions and their conjugates

Abstract

A fundamental theorem in discrete convex analysis states that a set function is M$^\natural$-concave if and only if its conjugate function is submodular. This paper gives a new proof to this fact.