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Distributed Set Cover Approximation: Primal-Dual with Optimal Locality

Authors Guy Even, Mohsen Ghaffari, Moti Medina

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Guy Even
  • Tel-Aviv University, Israel
Mohsen Ghaffari
  • ETH Zurich, Switzerland
Moti Medina
  • Ben-Gurion University, Israel

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Guy Even, Mohsen Ghaffari, and Moti Medina. Distributed Set Cover Approximation: Primal-Dual with Optimal Locality. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.DISC.2018.22

Abstract

This paper presents a deterministic distributed algorithm for computing an f(1+epsilon) approximation of the well-studied minimum set cover problem, for any constant epsilon>0, in O(log (f Delta)/log log (f Delta)) rounds. Here, f denotes the maximum element frequency and Delta denotes the cardinality of the largest set. This f(1+epsilon) approximation almost matches the f-approximation guarantee of standard centralized primal-dual algorithms, which is known to be essentially the best possible approximation for polynomial-time computations. The round complexity almost matches the Omega(log (Delta)/log log (Delta)) lower bound of Kuhn, Moscibroda, Wattenhofer [JACM'16], which holds for even f=2 and for any poly(log Delta) approximation. Our algorithm also gives an alternative way to reproduce the time-optimal 2(1+epsilon)-approximation of vertex cover, with round complexity O(log Delta/log log Delta), as presented by Bar-Yehuda, Censor-Hillel, and Schwartzman [PODC'17] for weighted vertex cover. Our method is quite different and it can be viewed as a locality-optimal way of performing primal-dual for the more general case of set cover. We note that the vertex cover algorithm of Bar-Yehuda et al. does not extend to set cover (when f >= 3).

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Distributed algorithms
Keywords
  • Distributed Algorithms
  • Approximation Algorithms
  • Set Cover
  • Vertex Cover

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