Average-case complexity of Maximum Weight Independent Set with random weights in bounded degree graphs

Finding the largest independent set in a graph is a notoriously difficult NP -complete combinatorial optimization problem. Unlike other NP-complete problems, it does not admit a constant factor approximation algorithm for general graphs. Furthermore, even for graphs with largest degree 3, no polynomial time approximation algorithm exists with a 1.0071-factor approximation guarantee. We consider the problem of finding maximum weight independent set in bounded degree graph, when the node weights are generated i.i.d. from a common distribution. For instance, we construct a PTAS (Polynomial-Time Approximation Scheme) for the case of exponentially distributed weights and arbitrary graphs with degree at most 3. We generalize the analysis to phase-type distributions (dense in the space of all distributions), and provide partial converse results, showing that even under a random cost assumption, it can be NP-hard to compute the MWIS of a graph with sufficiently large degree. We also show how the method can be used to derive lower bounds for the expected size of maximum independent set in large random regular graphs. Our algorithm, the cavity expansion, is based on combination of several ideas, including recent deterministic approximation algorithms for counting on graphs and local weak convergence/correlation decay methods.