PTAS for maximum weight independent set problem with random weights in bounded degree graphs

Abstract

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.