Core-based criterion for extreme supermodular functions

TL;DR

This paper introduces a core-based criterion for determining extremality of supermodular functions, linking their min-representation to core polytope vertices, and characterizes indecomposability within generalized permutohedra.

Contribution

It provides a necessary and sufficient condition for extremality of supermodular functions using core polytope vertices, advancing understanding of their structure and indecomposability.

Findings

Characterization of extremality via core vertices

Linear equation system for extremality condition

Comparison with existing extremality criteria

Abstract

We give a necessary and sufficient condition for extremality of a supermodular function based on its min-representation by means of (vertices of) the corresponding core polytope. The condition leads to solving a certain simple linear equation system determined by the combinatorial core structure. Our result allows us to characterize indecomposability in the class of generalized permutohedra. We provide an in-depth comparison between our result and the description of extremality in the supermodular/submodular cone achieved by other researchers.