Improved Local Computation Algorithms for Greedy Set Cover via Retroactive Updates

2026-03-24

AI Summary

This work proposes a novel local computation algorithm (LCA) for the set cover problem that significantly reduces query complexity while preserving approximation quality. The key innovation lies in an aggressive input sparsification strategy combined with a “backtrack-and-update” mechanism, which enables the algorithm to dynamically revise decisions made in earlier recursive calls. This enhances solution concentration and curtails redundant computation. Theoretical analysis demonstrates that the proposed method lowers the query complexity from Δ^{O(log Δ)} to f^{O(log Δ)}; moreover, when f = polylog Δ, the complexity is further improved to Δ^{O(log log Δ)}, substantially outperforming existing approaches.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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 $O(\log Δ)$-approximate set cover, developed by Grunau, Mitrović, Rubinfeld, and Vakilian [SODA '20], achieves query complexity of $Δ^{O(\log Δ)} \cdot f^{O(\log Δ\cdot (\log \log Δ+ \log \log f))}$, where $Δ$ 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 $f^{O(\log Δ)}$ queries. Specifically, for instances where $f = \text{poly} \log Δ$, our algorithm improves the query complexity from $Δ^{O(\log Δ)}$ to $Δ^{O(\log \log Δ)}$. 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.

Problem

• Local Computation Algorithm
• Set Cover
• Query Complexity
• Greedy Algorithm
• Retroactive Updates

Innovation

• Local Computation Algorithms
• Set Cover
• Retroactive Updates
• Query Complexity
• Sparsification