A Dynamic Algorithm for Weighted Submodular Cover Problem

TL;DR

This paper introduces a randomized dynamic algorithm for the weighted submodular cover problem, efficiently maintaining approximate solutions under element insertions and deletions with low query complexity.

Contribution

It presents the first dynamic algorithm achieving a bicriteria approximation for the weighted submodular cover problem with polylogarithmic query complexity per update.

Findings

Achieves a (1-O(ε), O(ε^{-1}))-bicriteria approximation.

Operates with polylogarithmic query complexity per update.

Handles dynamic insertions and deletions efficiently.

Abstract

We initiate the study of the submodular cover problem in dynamic setting where the elements of the ground set are inserted and deleted. In the classical submodular cover problem, we are given a monotone submodular function $f : 2^{V} \to \mathbb{R}^{\ge 0}$ and the goal is to obtain a set $S \subseteq V$ that minimizes the cost subject to the constraint $f(S) = f(V)$. This is a classical problem in computer science and generalizes the Set Cover problem, 2-Set Cover, and dominating set problem among others. We consider this problem in a dynamic setting where there are updates to our set $V$, in the form of insertions and deletions of elements from a ground set $\mathcal{V}$, and the goal is to maintain an approximately optimal solution with low query complexity per update. For this problem, we propose a randomized algorithm that, in expectation, obtains a $(1-O(\epsilon), O(\epsilon^{-1}))$-bicriteria approximation using polylogarithmic query complexity per update.