An Approximation Algorithm for Monotone Submodular Cost Allocation

TL;DR

This paper introduces a $k/2$-approximation algorithm for the monotone submodular cost allocation problem, matching the integrality gap and advancing understanding of approximation bounds for this class of problems.

Contribution

It provides a tight approximation ratio for the MSCA problem with monotone submodular functions using LP relaxation analysis.

Findings

The integrality gap of the LP relaxation is at most $k/2$.

A $k/2$-approximation algorithm is developed.

The lower bound on the integrality gap is nearly $k/2$ for fixed $k$.

Abstract

In this paper, we consider the minimum submodular cost allocation (MSCA) problem. The input of MSCA is $k$ non-negative submodular functions $f_1,f_2,\ldots,f_k$ on the ground set $N$ given by evaluation oracles, and the goal is to partition $N$ into $k$ (possibly empty) sets $S_1,S_2,\ldots,S_k$ so that $\sum_{i=1}^k f_i(S_i)$ is minimized. In this paper, we focus on the case when $f_1,f_2,\ldots,f_k$ are monotone, which coincides with the submodular facility location problem considered by Svitkina and Tardos. We show that the integrality gap of a natural LP-relaxation for MSCA with monotone submodular functions is at most $k/2$, yielding a $k/2$-approximation algorithm. We also prove a nearly matching lower bound: the integrality gap is at least $k/2-\epsilon$ for any constant $\epsilon>0$ when $k$ is fixed.