A Local Search Algorithm for the Min-Sum Submodular Cover Problem

TL;DR

This paper introduces a local search algorithm for the Min-Sum Submodular Cover problem, achieving near-optimal solutions efficiently under certain conditions, and demonstrates its practical advantages over greedy methods in experiments.

Contribution

It presents a local search algorithm with provable approximation guarantees for the Min-Sum Submodular Cover problem, extending previous greedy approaches and applicable to practical submodular functions.

Findings

Local search achieves a (4+ε)-approximation in polynomial time.

The algorithm outperforms greedy on small datasets.

Applicable to functions like set cover, matching, and facility location.

Abstract

We consider the problem of solving the Min-Sum Submodular Cover problem using local search. The Min-Sum Submodular Cover problem generalizes the NP-complete Min-Sum Set Cover problem, replacing the input set cover instance with a monotone submodular set function. A simple greedy algorithm achieves an approximation factor of 4, which is tight unless P=NP [Streeter and Golovin, NeurIPS, 2008]. We complement the greedy algorithm with analysis of a local search algorithm. Building on work of Munagala et al. [ICDT, 2005], we show that, using simple initialization, a straightforward local search algorithm achieves a $(4+\epsilon)$-approximate solution in time $O(n^3\log(n/\epsilon))$, provided that the monotone submodular set function is also second-order supermodular. Second-order supermodularity has been shown to hold for a number of submodular functions of practical interest, including functions associated with set cover, matching, and facility location. We present experiments on two special cases of Min-Sum Submodular Cover and find that the local search algorithm can outperform the greedy algorithm on small data sets.