Improved Local Computation Algorithm for Set Cover via Sparsification

TL;DR

This paper introduces an improved local computation algorithm for the set cover problem that uses sparsification and parallelization techniques to reduce query complexity and adaptivity, outperforming previous methods.

Contribution

It presents a novel LCA for set cover with better query complexity by leveraging sparsification and a non-revoking parallel algorithm, improving upon prior reductions.

Findings

Achieves lower query complexity than previous algorithms.

Introduces a sparsification technique for set cover.

Develops a non-revoking parallel set cover algorithm.

Abstract

We design a Local Computation Algorithm (LCA) for the set cover problem. Given a set system where each set has size at most $s$ and each element is contained in at most $t$ sets, the algorithm reports whether a given set is in some fixed set cover whose expected size is $O(\log{s})$ times the minimum fractional set cover value. Our algorithm requires $s^{O(\log{s})} t^{O(\log{s} \cdot (\log \log{s} + \log \log{t}))}$ queries. This result improves upon the application of the reduction of [Parnas and Ron, TCS'07] on the result of [Kuhn et al., SODA'06], which leads to a query complexity of $(st)^{O(\log{s} \cdot \log{t})}$. To obtain this result, we design a parallel set cover algorithm that admits an efficient simulation in the LCA model by using a sparsification technique introduced in [Ghaffari and Uitto, SODA'19] for the maximal independent set problem. The parallel algorithm adds a random subset of the sets to the solution in a style similar to the PRAM algorithm of [Berger et al., FOCS'89]. However, our algorithm differs in the way that it never revokes its decisions, which results in a fewer number of adaptive rounds. This requires a novel approximation analysis which might be of independent interest.