Abstract
We present a deterministic distributed 2-approximation algorithm for the Minimum Weight Vertex Cover problem in the CONGEST model whose round complexity is \(O(\log n\log \varDelta / \log ^2\log \varDelta )\). This improves over the currently best known deterministic 2-approximation implied by [KVY94]. Our solution generalizes the \((2+\epsilon )\)-approximation algorithm of [BCS17], improving the dependency on \(\epsilon ^{-1}\) from linear to logarithmic. In addition, for every \(\epsilon =(\log \varDelta )^{-c}\), where \(c\ge 1\) is a constant, our algorithm computes a \(\left( 2+\epsilon \right) \)-approximation in \(O(\log {\varDelta }/\log \log {\varDelta })\) rounds (which is asymptotically optimal).
R. Ben-Basat—This work was partially sponsored by the Technion-HPI research school.
K. Kawarabayashi and G. Schwartzman—This work was supported by JST ERATO Grant Number JPMJER1201, Japan.
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Notes
- 1.
In the CONGEST model vertices have distinct IDs (that are polynomial in |V|), however, as in [BCS17], our algorithm works also in the case of anonymous vertices.
- 2.
All logarithms are base 2 unless the basis is written explicitly.
- 3.
The actual result is stated as a \((2+\epsilon )\)-approximation in \(O(\log \epsilon ^{-1} \log n)\) rounds, from which we infer a 2-approximation by setting \(\epsilon =1/nW\).
- 4.
A polynomial upper bound of \(\varDelta ^{O(1)}\) would yield the same asymptotic bound on the number of rounds.
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Ben-Basat, R., Even, G., Kawarabayashi, Ki., Schwartzman, G. (2018). A Deterministic Distributed 2-Approximation for Weighted Vertex Cover in \(O(\log N\log \varDelta /\log ^2\log \varDelta )\) Rounds. In: Lotker, Z., Patt-Shamir, B. (eds) Structural Information and Communication Complexity. SIROCCO 2018. Lecture Notes in Computer Science(), vol 11085. Springer, Cham. https://doi.org/10.1007/978-3-030-01325-7_21
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