Improved Local Computation Algorithms for Greedy Set Cover via Retroactive Updates

TL;DR

This paper introduces a new local computation algorithm for the set cover problem that significantly reduces query complexity by using retroactive updates to correct previous decisions, leading to more efficient solutions.

Contribution

The authors develop a novel LCA technique employing retroactive updates to improve query complexity for set cover approximation.

Findings

Query complexity improved from $ ext{poly}( ext{log} \Delta)$ to $ ext{poly}( ext{log} ext{log} \Delta)$ for certain instances.

Retroactive updates enable stronger concentration guarantees and sparser algorithm execution.

The new method outperforms previous state-of-the-art LCAs for the set cover problem.

Abstract

In this work, we focus on designing an efficient Local Computation Algorithm (LCA) for the set cover problem, which is a core optimization task. The state-of-the-art LCA for computing $O(\log \Delta)$-approximate set cover, developed by Grunau, Mitrovi\'c, Rubinfeld, and Vakilian [SODA '20], achieves query complexity of $\Delta^{O(\log \Delta)} \cdot f^{O(\log \Delta \cdot (\log \log \Delta + \log \log f))}$, where $\Delta$ is the maximum set size, and $f$ is the maximum frequency of any element in sets. We present a new LCA that solves this problem using $f^{O(\log \Delta)}$ queries. Specifically, for instances where $f = \text{poly} \log \Delta$, our algorithm improves the query complexity from $\Delta^{O(\log \Delta)}$ to $\Delta^{O(\log \log \Delta)}$. Our central technical contribution in designing LCAs is to aggressively sparsify the input instance but to allow for \emph{retroactive updates}. Namely, our main LCA sometimes ``corrects'' decisions it made in the previous recursive LCA calls. It enables us to achieve stronger concentration guarantees, which in turn allows for more efficient and ``sparser'' LCA execution. We believe that this technique will be of independent interest.