Randomized Greedy Algorithms for Independent Sets and Matchings in Regular Graphs: Exact Results and Finite Girth Corrections
Author(s)
Gamarnik, David; Goldberg, David A.
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We derive new results for the performance of a simple greedy algorithm for finding large independent sets and matchings in constant degree regular graphs. We show that for r-regular graphs with n nodes and girth at least g, the algorithm finds an independent set of expected cardinality
f(r)n-O((r-1)g/2/g/2!n) [(f(r)n-O (r-1) superscript g over 2 / g over 2 ! n)], where f(r) is a function which we explicitly compute. A similar result is established
for matchings. Our results imply improved bounds for the size of the largest independent set
in these graphs, and provide the first results of this type for matchings. As an implication we show
that the greedy algorithm returns a nearly perfect matching when both the degree r and girth g are
large. Furthermore, we show that the cardinality of independent sets and matchings produced by
the greedy algorithm in arbitrary bounded degree graphs is concentrated around the mean. Finally,
we analyze the performance of the greedy algorithm for the case of random i.i.d. weighted independent
sets and matchings, and obtain a remarkably simple expression for the limiting expected values
produced by the algorithm. In fact, all the other results are obtained as straightforward corollaries
from the results for the weighted case.
Date issued
2009-06Department
Massachusetts Institute of Technology. Operations Research Center; Sloan School of ManagementJournal
Combinatorics, Probability and Computing
Publisher
Cambridge University Press
Citation
Gamarnik, David, and David A. Goldberg. “Randomized Greedy Algorithms for Independent Sets and Matchings in Regular Graphs: Exact Results and Finite Girth Corrections.” Combinatorics, Probability and Computing 19.01 (2009) : 61. Copyright © Cambridge University Press 2009
Version: Author's final manuscript
ISSN
0963-5483