Polynomial algorithm for $k$-partition minimization of monotone submodular function

TL;DR

This paper presents a polynomial-time algorithm for the $k$-partition minimization problem of symmetric submodular functions, extending Queyranne's algorithm for 2-partitions to arbitrary fixed $k$ with complexity $O(n^{3(k-1)})

Contribution

It introduces a recursive algorithm leveraging Queyranne's 2-partition minimization to efficiently solve the $k$-partition problem for symmetric submodular functions.

Findings

Algorithm has complexity $O(n^{3(k-1)})$ for fixed $k$

Extends Queyranne's 2-partition algorithm to $k$-partitions

Provides a polynomial-time solution for fixed $k$

Abstract

For a fixed $k$, this study considers $k$-partition minimization of submodular system $(V, f)$ with a finite set $V$ and symmetric submodular function $f: 2^{V} \mapsto \mathbb{R}$. Our algorithm uses the Queyranne's (1998) algorithm for 2-partition minimization which arises at each step of the recursive decomposition of subsets of the original $k$-partition minimization. We show that the computational complexity of this minimizer is $O(n^{3(k-1)})$.