Paper Info
Title
Randomized greedy algorithms for independent sets and matchings in regular graphs: Exact results and finite girth corrections
Abstract
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.
Year
DOI
Venue
2008
10.1017/S0963548309990186
Clinical Orthopaedics and Related Research
Keywords
DocType
Volume
finite girth correction,expected cardinality,weighted case,exact result,greedy algorithm,large independent set,simple greedy algorithm,largest independent set,simple expression,regular graph,randomized greedy algorithm,girth g,independent set,weighted independent set
Journal
19
Issue
ISSN
Citations 
1
0963-5483
9
PageRank 
References 
Authors
0.62
20
2
Name
Order
Citations
PageRank
DAVID GAMARNIK164161.04
David A. Goldberg2212.32