Optimal Distributed Weighted Set Cover Approximation

TL;DR

This paper introduces a time-optimal deterministic distributed algorithm for approximating minimum weight vertex cover in hypergraphs, achieving an approximation factor of (f+ε) with a running time independent of vertex weights or count.

Contribution

It presents the first distributed algorithm with a running time independent of vertex weights or the number of vertices, achieving optimal time complexity for hypergraph set cover approximation.

Findings

Runs in O(log Δ / log log Δ) rounds in the CONGEST model.

Achieves approximation factor of (f+ε).

Improves over previous algorithms with dependence on vertex weights.

Abstract

We present a time-optimal deterministic distributed algorithm for approximating a minimum weight vertex cover in hypergraphs of rank $f$. This problem is equivalent to the Minimum Weight Set Cover Problem in which the frequency of every element is bounded by $f$. The approximation factor of our algorithm is $(f+\epsilon)$. Let $\Delta$ denote the maximum degree in the hypergraph. Our algorithm runs in the CONGEST model and requires $O(\log{\Delta} / \log \log \Delta)$ rounds, for constants $\epsilon \in (0,1]$ and $f\in\mathbb N^+$. This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights or the number of vertices. Thus adding another member to the exclusive family of \emph{provably optimal} distributed algorithms. For constant values of $f$ and $\epsilon$, our algorithm improves over the $(f+\epsilon)$-approximation algorithm of \cite{KuhnMW06} whose running time is $O(\log \Delta + \log W)$, where $W$ is the ratio between the largest and smallest vertex weights in the graph.