A Combinatorial, Strongly Polynomial-Time Algorithm for Minimizing Submodular Functions

TL;DR

This paper introduces the first combinatorial polynomial-time algorithm for minimizing submodular functions, solving a long-standing open problem and providing both scaled and strongly polynomial-time versions.

Contribution

It presents a novel combinatorial algorithm for submodular function minimization with a strongly polynomial-time variant, answering a question from 1981.

Findings

Algorithm runs in polynomial time relative to set size and function value

A strongly polynomial-time algorithm is developed, independent of function values

First combinatorial polynomial-time algorithm for this problem

Abstract

This paper presents the first combinatorial polynomial-time algorithm for minimizing submodular set functions, answering an open question posed in 1981 by Grotschel, Lovasz, and Schrijver. The algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter. The resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the largest length of the function value. The paper also presents a strongly polynomial-time version that runs in time bounded by a polynomial in the size of the underlying set independent of the function value.