Randomized Greedy Algorithms for Independent Sets and Matchings in Regular Graphs: Exact Results and Finite Girth Corrections
Name
Goldberg_Gamarnik_CPC2009.pdf
Size
251.85 KB
Format
Adobe PDF
Checksum (MD5)
5d42c9d81b81a721583a2462c7759f2a
Author(s)
Gamarnik, David
Goldberg, David A.
Date Issued
June 2009
Journal
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
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((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.
MIT Department
Massachusetts Institute of Technology. Operations Research Center
Sloan School of Management
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike 3.0
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1017/s0963548309990186
