Basic facts concerning supermodular functions

TL;DR

This paper reviews fundamental properties of supermodular set functions, focusing on operations that preserve supermodularity and extremality, including transformations, projections, and extensions within the cone of such functions.

Contribution

It characterizes operations on supermodular functions that maintain supermodularity and extremality, providing insights into transformations, projections, and extensions of these functions.

Findings

Identifies operations preserving supermodularity.

Describes transformations that preserve extremality.

Analyzes projections and extensions maintaining supermodularity.

Abstract

Elementary facts and observations on the cone of supermodular set functions are recalled. The manuscript deals with such operations with set functions which preserve supermodularity and the emphasis is put on those such operations which even preserve extremality (of a supermodular function). These involve a few self-transformations of the cone of supermodular set functions. Moreover, projections to the (less-dimensional) linear space of set functions for a subset of the variable set are discussed. Finally, several extensions to the (more-dimensional) linear space of set functions for a superset of the variable set are shown to be both preserving supermodularity and extremality.