Bicriteria Approximation Algorithms for the Submodular Cover Problem

TL;DR

This paper introduces new scalable algorithms for various forms of the Submodular Cover problem, achieving near-optimal solutions efficiently and demonstrating effectiveness in data summarization and graph cut applications.

Contribution

It presents the first algorithms for regularized SCP and near-feasible solutions for general SCP, with improved scalability for monotone SCP.

Findings

Scalable algorithm for monotone SCP with near-greedy guarantees

First algorithms for regularized SCP

Effective performance in data summarization and graph cut tasks

Abstract

In this paper, we consider the optimization problem Submodular Cover (SCP), which is to find a minimum cardinality subset of a finite universe $U$ such that the value of a submodular function $f$ is above an input threshold $\tau$. In particular, we consider several variants of SCP including the general case, the case where $f$ is additionally assumed to be monotone, and finally the case where $f$ is a regularized monotone submodular function. Our most significant contributions are that: (i) We propose a scalable algorithm for monotone SCP that achieves nearly the same approximation guarantees as the standard greedy algorithm in significantly faster time; (ii) We are the first to develop an algorithm for general SCP that achieves a solution arbitrarily close to being feasible; and finally (iii) we are the first to develop algorithms for regularized SCP. Our algorithms are then demonstrated to be effective in an extensive experimental section on data summarization and graph cut, two applications of SCP.