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Randomized Greedy Algorithms for Independent Sets and Matchings in Regular Graphs: Exact Results and Finite Girth Corrections

Published online by Cambridge University Press:  22 June 2009

DAVID GAMARNIK and
DAVID A. GOLDBERG
DAVID GAMARNIK
Affiliation:
Operations Research Center and Sloan School of Management, MIT, Cambridge, MA 02139, USA (e-mail: gamarnik@mit.edu)
DAVID A. GOLDBERG
Affiliation:
Operations Research Center, MIT, Cambridge, MA 02139, USA (e-mail: dag3141@mit.edu)
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Abstract

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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\[ f(r)n - O\biggl(\frac{(r-1)^{\frac{g}{2}}}{ \frac{g}{2}!} n\biggr), \]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 analyse 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.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 1 , January 2010, pp. 61 - 85
Copyright
Copyright © Cambridge University Press 2009

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