Minimizing a sum of submodular functions

TL;DR

This paper presents a novel approach to minimize sums of submodular functions by casting the problem as a submodular flow problem, leveraging specialized exchange capacity routines, and improving complexity for cardinality-dependent terms.

Contribution

It introduces a submodular flow formulation for sum-of-submodular functions and enhances Iwata's capacity scaling algorithm for specific term types.

Findings

Efficiently minimizes sum of submodular functions using submodular flow formulation.

Improves complexity of existing algorithms for cardinality-dependent submodular terms.

Demonstrates practical benefits of specialized exchange capacity routines.

Abstract

We consider the problem of minimizing a function represented as a sum of submodular terms. We assume each term allows an efficient computation of {\em exchange capacities}. This holds, for example, for terms depending on a small number of variables, or for certain cardinality-dependent terms. A naive application of submodular minimization algorithms would not exploit the existence of specialized exchange capacity subroutines for individual terms. To overcome this, we cast the problem as a {\em submodular flow} (SF) problem in an auxiliary graph, and show that applying most existing SF algorithms would rely only on these subroutines. We then explore in more detail Iwata's capacity scaling approach for submodular flows (Math. Programming, 76(2):299--308, 1997). In particular, we show how to improve its complexity in the case when the function contains cardinality-dependent terms.