Submodular functions and convexity

Abstract

This chapter describes the essence of discrete convex analysis in a compact way with the help of the theory of submodular functions and the ordinary convex analysis. Only polyhedral convex functions are considered. Historical notes about developments in discrete convex analysis are given. The chapter reviews the theory of ordinary convex analysis, focusing on locally polyhedral convex functions, and also gives definitions of some concepts about discrete convexity. There are several operations on base polyhedra that are closed within the class of base polyhedra. Similar operations such as a truncation and its dual can be adapted to define the corresponding operations on M-convex functions. The chapter shows a one-to-one conjugacy correspondence between the set of integer-valued domain-integral L-convex functions and that of integer-valued domain-integral M-convex functions. M-convex functions have the exchange property. The exchange property is adopted as the defining axiom for M-convex functions in Murota's discrete convex analysis. Proximity theorems are concerned with solutions of relaxed or restricted problems modified from an original one and show how close to an optimal solution of the original problem the approximate solutions are.