Monotone Submodular Multiway Partition

TL;DR

This paper studies the problem of partitioning a set to minimize a monotone submodular function, providing approximation algorithms and hardness results, and introduces a special case related to graph coverage with tighter bounds.

Contribution

It presents the first approximation algorithms for monotone submodular multiway partition and its special case, graph coverage, along with hardness results under the Unique Games Conjecture.

Findings

Monotone submodular multiway partition admits a 4/3-approximation.

No $(10/9 - \epsilon)$-approximation exists for the general problem.

Graph coverage multiway partition admits a 1.125-approximation and is hard to approximate within 1.00074 assuming UGC.

Abstract

In submodular multiway partition (SUB-MP), the input is a non-negative submodular function $f:2^V \rightarrow \mathbb{R}_{\ge 0}$ given by an evaluation oracle along with $k$ terminals $t_1, t_2, \ldots, t_k\in V$. The goal is to find a partition $V_1, V_2, \ldots, V_k$ of $V$ with $t_i\in V_i$ for every $i\in [k]$ in order to minimize $\sum_{i=1}^k f(V_i)$. In this work, we focus on SUB-MP when the input function is monotone (termed MONO-SUB-MP). MONO-SUB-MP formulates partitioning problems over several interesting structures -- e.g., matrices, matroids, graphs, and hypergraphs. MONO-SUB-MP is NP-hard since the graph multiway cut problem can be cast as a special case. We investigate the approximability of MONO-SUB-MP: we show that it admits a $4/3$-approximation and does not admit a $(10/9-\epsilon)$-approximation for every constant $\epsilon>0$. Next, we study a special case of MONO-SUB-MP where the monotone submodular function of interest is the coverage function of an input graph, termed GRAPH-COVERAGE-MP. GRAPH-COVERAGE-MP is equivalent to the classic multiway cut problem for the purposes of exact optimization. We show that GRAPH-COVERAGE-MP admits a $1.125$-approximation and does not admit a $(1.00074-\epsilon)$-approximation for every constant $\epsilon>0$ assuming the Unique Games Conjecture. These results separate GRAPH-COVERAGE-MP from graph multiway cut in terms of approximability.