PTAS for maximum weight independent set problem with random weights in bounded degree graphs
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Gamarnik_PTAS for maximum.pdf
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Author(s)
Gamarnik, David
Goldberg, David A.
Weber, Theophane G.
Date Issued
January 2010
Journal
Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms
Publisher
Association for Computing Machinery
Citation
Gamarnik, David, David Goldberg, and Theophane Weber. “PTAS for maximum weight independent set problem with random weights in bounded degree graphs.” Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms (2010): 268–278. © 2010 Society for Industrial and Applied Mathematics
Version
Final published version
Abstract
Finding the largest independent set in a graph is a notoriously difficult NP-complete combinatorial optimization problem. Moreover, even for graphs with largest degree 3, no polynomial time approximation algorithm exists with a 1.0071-factor approximation guarantee, unless P = NP [BK98]. We consider the related problem of finding the
maximum weight independent set in a bounded degree graph, when the node weights are generated i.i.d. from a common distribution. Surprisingly, we discover that the problem becomes tractable for certain distributions. Specifically, we construct a randomized
PTAS (Polynomial-Time Approximation Scheme) for the case of exponentially distributed weights and arbitrary graphs with degree at most 3. We extend our result to graphs with larger constant degrees but for distributions which are mixtures of exponential distributions. At the same time, we prove that no PTAS exists for computing the expected size of the maximum weight independent set in the case of exponentially distributed
weights for graphs with sufficiently large constant degree, unless P=NP. Our algorithm, cavity expansion, is new and is based on the combination of several powerful ideas, including recent deterministic approximation algorithms for counting on graphs and local weak convergence/correlation decay methods.
MIT Department
Massachusetts Institute of Technology. Operations Research Center
Sloan School of Management
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
http://www.siam.org/proceedings/soda/2010/soda10.php
