Dynamic Set Cover with Worst-Case Recourse

TL;DR

This paper introduces a transformation for dynamic set cover algorithms that achieves low worst-case recourse and update time while maintaining a good approximation ratio, addressing a key open problem in the field.

Contribution

It provides a novel transformation that converts existing algorithms into ones with low worst-case recourse and update time for dynamic set cover.

Findings

Achieves low worst-case recourse in dynamic set cover

Provides a transformation that improves existing algorithms

Balances approximation ratio, update time, and recourse effectively

Abstract

In the dynamic set cover (SC) problem, the input is a dynamic universe of at most $n$ elements and a fixed collection of $m$ sets, where each element belongs to at most $f$ sets and each set has cost in $[1/C, 1]$. The objective is to efficiently maintain an approximate minimum SC under element updates; efficiency is primarily measured by the update time, but another important parameter is the recourse (number of changes to the solution per update). Ideally, one would like to achieve low worst-case bounds on both update time and recourse. One can achieve approximation $(1+\epsilon)\ln n$ (greedy-based) or $(1+\epsilon)f$ (primal-dual-based) with worst-case update time $O(f\log n)$ (ignoring $\epsilon$ dependencies). However, despite a large body of work, no algorithm with low update time (even amortized) and nontrivial worst-case recourse is known, even for unweighted instances ($C = 1$)! We remedy this by providing a transformation that, given as a black-box a SC algorithm with approximation $\alpha$ and update time $T$, returns a set cover algorithm with approximation $(2 + \epsilon)\alpha$, update time $O(T + \alpha C)$, and worst-case recourse $O(\alpha C)$. Our main results are obtained by leveraging this transformation for constant $C$:...