A near-Optimal Sublinear-Time Algorithm for Approximating the Minimum Vertex Cover Size
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Author(s)
Onak, Krzysztof
Ron, Dana
Rosen, Michal
Rubinfeld, Ronitt
Date Issued
2012
Journal
Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '12)
Publisher
Association for Computing Machinery (ACM)
Citation
Krzysztof Onak, Dana Ron, Michal Rosen, and Ronitt Rubinfeld. 2012. A near-optimal sublinear-time algorithm for approximating the minimum vertex cover size. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '12).
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Author's final manuscript
Abstract
We give a nearly optimal sublinear-time algorithm for approximating the size of a minimum vertex cover in a graph G. The algorithm may query the degree deg(v) of any vertex v of its choice, and for each 1 ≤ i ≤ deg(v), it may ask for the i[superscript th] neighbor of v. Letting VCopt(G) denote the minimum size of vertex cover in G, the algorithm outputs, with high constant success probability, an estimate [EQUATION] such that [EQUATION], where ε is a given additive approximation parameter. We refer to such an estimate as a (2, ε)-estimate. The query complexity and running time of the algorithm are Õ([EQUATION] · poly(1/ε)), where d denotes the average vertex degree in the graph. The best previously known sublinear algorithm, of Yoshida et al. (STOC 2009), has query complexity and running time O(d[superscript 4]/ε[superscript 2]), where d is the maximum degree in the graph. Given the lower bound of Ω(d) (for constant ε) for obtaining such an estimate (with any constant multiplicative factor) due to Parnas and Ron (TCS 2007), our result is nearly optimal.
In the case that the graph is dense, that is, the number of edges is Θ(n[superscript 2]), we consider another model, in which the algorithm may ask, for any pair of vertices u and v, whether there is an edge between u and v. We show how to adapt the algorithm that uses neighbor queries to this model and obtain an algorithm that outputs a (2, ε)-estimate of the size of a minimum vertex cover whose query complexity and running time are Õ(n) · poly(1/ε).
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
http://dl.acm.org/citation.cfm?id=2095204
