Approximately counting up to four (extended abstract)
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Non-approximability results for optimization problems on bounded degree instances
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
The ζ (2) limit in the random assignment problem
Random Structures & Algorithms
Clique is hard to approximate within n1-
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
On Some Tighter Inapproximability Results
On Some Tighter Inapproximability Results
Random Structures & Algorithms
Counting independent sets up to the tree threshold
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Simple deterministic approximation algorithms for counting matchings
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Correlation decay and deterministic FPTAS for counting list-colorings of a graph
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Maximum matching in sparse random graphs
SFCS '81 Proceedings of the 22nd Annual Symposium on Foundations of Computer Science
Combinatorics, Probability and Computing
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Finding the largest independent set in a graph is a notoriously difficult N P-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 = N P [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.