Attempting the impossible: enumerating extremal submodular functions for n=6

TL;DR

This paper advances the enumeration of extremal submodular functions on a six-element set, employing novel computational methods to estimate their total count and analyze their structural properties.

Contribution

It introduces two new ordering strategies that significantly improve enumeration efficiency and provides the first estimate of the total number of extremal submodular functions for six elements.

Findings

Enumerated 360 billion extremal submodular functions for n=6

Estimated total number between 1,000 and 10,000 times the enumerated count

Developed improved polyhedral geometry tools and methods

Abstract

Enumerating the extremal submodular functions defined on subsets of a fixed base set has only been done for base sets up to five elements. This paper reports the results of attempting to generate all such functions on a six-element base set. Using improved tools from polyhedral geometry, we have computed 360 billion of them, and provide the first reasonable estimate of their total number, which is expected to be between 1,000 and 10,000 times this number. The applied Double Description and Adjacency Decomposition methods require an insertion order of the defining inequalities. We introduce two novel orders, which speed up the computations significantly, and provide additional insight into the highly symmetric structure of submodular functions. We also present an improvement to the combinatorial test used as part of the Double Description method, and use statistical analyses to estimate the degeneracy of the polyhedral cone used to describe these functions. The statistical results also highlight the limitations of the applied methods.