# Brief Announcement: Optimal Distributed Covering Algorithms

Ran Ben Basat, Guy Even, Ken Ichi Kawarabayashi, Gregory Schwartzman

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

## Abstract

We present a time-optimal deterministic distributed algorithm for approximating a minimum weight vertex cover in hypergraphs of rank f . This problem is equivalent to the Minimum Weight Set Cover problem in which the frequency of every element is bounded by f . The approximation factor of our algorithm is (f +ϵ). Let δ denote the maximum degree in the hypergraph. Our algorithm runs in the congest model and requiresO(log δ/log log δ) rounds, for constants ϵ ϵ (0, 1] and f ϵ ℕ+. This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights nor the number of vertices. Thus adding another member to the exclusive family of provably optimal distributed algorithms. For constant values of f and ϵ, our algorithm improves over the (f + ϵ)-approximation algorithm of  whose running time is O(log δ + logW), whereW is the ratio between the largest and smallest vertexweights in the graph. Our algorithm also achieves an f -approximation for the problem in O(f logn) rounds, improving over the classical result of  that achieves a running time of O(f log2 n). Finally, forweighted vertex cover (f = 2) our algorithm achieves a deterministic running time of O(logn), matching the randomized previously best result of . We also show that integer covering-programs can be reduced to the Minimum Weight Set Cover problem in the distributed setting. This allows us to achieve an (f +ϵ)-approximate integral solution in (equation presented) rounds, where f bounds the number of variables in a constraint, δ bounds the number of constraints a variable appears in, and M = max {1, 1/amin}, where amin is the smallest normalized constraint coefficient. This significantly improves over the results of  for the integral case, which achieves the same guarantees in O (ϵ-4 · f4 · log f · log(M · δ) rounds.

Original language English PODC 2019 - Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing 104-106 3 9781450362177 https://doi.org/10.1145/3293611.3331577 Published - 16 Jul 2019 38th ACM Symposium on Principles of Distributed Computing, PODC 2019 - Toronto, CanadaDuration: 29 Jul 2019 → 2 Aug 2019

### Publication series

Name Proceedings of the Annual ACM Symposium on Principles of Distributed Computing

### Conference

Conference 38th ACM Symposium on Principles of Distributed Computing, PODC 2019 Canada Toronto 29/07/19 → 2/08/19

## Keywords

• Distributed computing
• Graph algorithms
• Primal dual
• Set cover
• Vertex cover

## All Science Journal Classification (ASJC) codes

• Software
• Hardware and Architecture
• Computer Networks and Communications

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