Approximating Submodular Matroid-Constrained Partitioning

TL;DR

This paper studies the problem of partitioning a set to minimize a submodular function under matroid constraints, unifying fixed terminal and global settings, and advances approximation algorithms especially for symmetric submodular functions.

Contribution

It introduces a generalized submodular matroid-constrained partition problem, unifying previous settings, and improves approximation results for symmetric, monotone, and general submodular functions.

Findings

Achieved state-of-the-art approximation for symmetric submodular functions.

Extended approximation techniques to monotone and general submodular functions.

Unified fixed terminal and global partitioning settings under matroid constraints.

Abstract

The submodular partitioning problem asks to minimize, over all partitions $P$ of a ground set $V$, the sum of a given submodular function $f$ over the parts of $P$. The problem has seen considerable work in approximability, as it encompasses multiterminal cuts on graphs, $k$-cuts on hypergraphs, and elementary linear algebra problems such as matrix multiway partitioning. This research has been divided between the fixed terminal setting, where we are given a set of terminals that must be separated by $P$, and the global setting, where the only constraint is the size of the partition. We investigate a generalization that unifies these two settings: minimum submodular matroid-constrained partition. In this problem, we are additionally given a matroid over the ground set and seek to find a partition $P$ in which there exists some basis that is separated by $P$. We explore the approximability of this problem and its variants, reaching the state of the art for the special case of symmetric submodular functions, and provide results for monotone and general submodular functions as well.