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Title: One-Way Communication Complexity of Minimum Vertex Cover in General Graphs
We study the communication complexity of the Minimum Vertex Cover (MVC) problem on general graphs within the k-party one-way communication model. Edges of an arbitrary n-vertex graph are distributed among k parties. The objective is for the parties to collectively find a small vertex cover of the graph while adhering to a communication protocol where each party sequentially sends a message to the next until the last party outputs a valid vertex cover of the whole graph. We are particularly interested in the trade-off between the size of the messages sent and the approximation ratio of the output solution. It is straightforward to see that any constant approximation protocol for MVC requires communicating Ω(n) bits. Additionally, there exists a trivial 2-approximation protocol where the parties collectively find a maximal matching of the graph greedily and return the subset of vertices matched. This raises a natural question: What is the best approximation ratio achievable using optimal communication of O(n)? We design a protocol with an approximation ratio of (2-2^{-k+1}+ε) and O(n) communication for any desirably small constant ε > 0, which is strictly better than 2 for any constant number of parties. Moreover, we show that achieving an approximation ratio smaller than 3/2 for the two-party case requires n^{1 + Ω(1/lg lg n)} communication, thereby establishing the tightness of our protocol for two parties. A notable aspect of our protocol is that no edges are communicated between the parties. Instead, for any 1 ≤ i < k, the i-th party only communicates a constant number of vertex covers for all edges assigned to the first i parties. An interesting consequence is that the communication cost of our protocol is O(n) bits, as opposed to the typical Ω(nlog n) bits required for many graph problems, such as maximum matching, where protocols commonly involve communicating edges.  more » « less
Award ID(s):
2335187
PAR ID:
10650553
Author(s) / Creator(s):
; ;
Editor(s):
Censor-Hillel, Keren; Grandoni, Fabrizio; Ouaknine, Joel; Puppis, Gabriele
Publisher / Repository:
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Date Published:
Volume:
334
ISSN:
1868-8969
Page Range / eLocation ID:
66:1-66:19
Subject(s) / Keyword(s):
Communication Complexity Minimum Vertex Cover Theory of computation → Communication complexity
Format(s):
Medium: X Size: 19 pages; 1004919 bytes Other: application/pdf
Size(s):
19 pages 1004919 bytes
Sponsoring Org:
National Science Foundation
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