The Expressive Power of Binary Submodular Functions

TL;DR

This paper investigates the expressive power of binary submodular functions, revealing limitations in their ability to represent all Boolean submodular functions and characterizing specific cases where they can be used for efficient minimization.

Contribution

It provides a negative answer to whether all Boolean submodular functions can be decomposed into binary submodular functions, and characterizes which functions can be expressed this way.

Findings

Not all Boolean submodular functions can be decomposed into binary submodular functions.

Identifies a class of submodular functions of arbitrary arity that can be expressed by binary submodular functions.

Establishes limitations of expressibility reductions for minimising arbitrary submodular functions.

Abstract

It has previously been an open problem whether all Boolean submodular functions can be decomposed into a sum of binary submodular functions over a possibly larger set of variables. This problem has been considered within several different contexts in computer science, including computer vision, artificial intelligence, and pseudo-Boolean optimisation. Using a connection between the expressive power of valued constraints and certain algebraic properties of functions, we answer this question negatively. Our results have several corollaries. First, we characterise precisely which submodular functions of arity 4 can be expressed by binary submodular functions. Next, we identify a novel class of submodular functions of arbitrary arities which can be expressed by binary submodular functions, and therefore minimised efficiently using a so-called expressibility reduction to the Min-Cut problem. More importantly, our results imply limitations on this kind of reduction and establish for the first time that it cannot be used in general to minimise arbitrary submodular functions. Finally, we refute a conjecture of Promislow and Young on the structure of the extreme rays of the cone of Boolean submodular functions.