Cones of elementary imsets and supermodular functions: a review and some new results

TL;DR

This paper reviews the geometric structure of elementary imsets and supermodular functions, and presents new findings on cone extremities, faces, relations, and computational aspects related to Markov bases.

Contribution

It provides a geometric perspective on imsets and introduces new results on cone structure, relations, and computational methods in this framework.

Findings

Characterization of extreme rays of supermodular cone

Identification of faces of the cones

Computational results on Markov bases

Abstract

In this paper we give a review of the method of imsets introduced by Studeny (2005) from a geometric point of view. Elementary imsets span a polyhedral cone and its dual cone is the cone of supermodular functions. We review basic facts on the structure of these cones. Then we derive some new results on the following topics: i) extreme rays of the cone of standardized supermodular functions, ii) faces of the cones, iii) small relations among elementary imsets, and iv) some computational results on Markov basis for the toric ideal defined by elementary imsets.