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Improved Local Computation Algorithms for Greedy Set Cover via Retroactive Updates

Slobodan Mitrović, Srikkanth Ramachandran, Ronitt Rubinfeld, Mihir Singhal

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.

Improved Local Computation Algorithms for Greedy Set Cover via Retroactive Updates

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 -approximate set cover, developed by Grunau, Mitrović, Rubinfeld, and Vakilian [SODA '20], achieves query complexity of , where is the maximum set size, and is the maximum frequency of any element in sets. We present a new LCA that solves this problem using queries. Specifically, for instances where , our algorithm improves the query complexity from to . 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.
Paper Structure (72 sections, 32 theorems, 42 equations, 11 algorithms)

This paper contains 72 sections, 32 theorems, 42 equations, 11 algorithms.

Table of Contents

  1. Introduction
  2. Local simulation of distributed algorithms.
  3. Open problems.
  4. Overview of our result and approach
  5. Organization of the paper.
  6. Starting point: Fractional Set Cover algorithm
  7. Details of \ref{['gmrv20:alg']}.
  8. Introducing slack in thresholds and pruning (\ref{['sec:warmup']})
  9. Observation 1.
  10. Why is Observation 1 not sufficient?
  11. Observation 2.
  12. A useful query recurrence
  13. Weaker invariants -- addressing \ref{['challenge:failure-prob']}
  14. Retroactive updates -- addressing \ref{['challenge:lca-design']}
  15. Modifying fractional to integral (\ref{['sec:frac-to-integral']})
  16. ...and 57 more sections

Key Result

Theorem 1.1

There exists an $\mathsf{LCA}$ for $\mathsf{SetCover}$ that outputs a randomized cover which is $O(\log \Delta)$-approximate in expectation while using $2^{O(\log \Delta \cdot \log f)}$ probes, where $\Delta$ is the maximum set size and $f$ the maximum element frequency.

Theorems & Definitions (59)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1: $\mathsf{SetCover}$
  • Definition 2.2: Local Computation Algorithm ($\mathsf{LCA}$)
  • Theorem 2.3: Chernoff bound
  • Corollary 2.4
  • proof
  • Lemma 3.0
  • Theorem 3.1
  • Lemma 3.2
  • ...and 49 more