Weighted Matchings via Unweighted Augmentations

Abstract

We design a generic method to reduce the task of finding weighted matchings to that of finding short augmenting paths in unweighted graphs. This method enables us to provide efficient implementations for approximating weighted matchings in the massively parallel computation (MPC) model and in the streaming model. For the MPC and the multi-pass streaming model, we show that any algorithm computing a (1-δ)-approximate unweighted matching in bipartite graphs can be translated into an algorithm that computes a (1-(ε(δ))-approximate maximum weighted matching. Furthermore, this translation incurs only a constant factor (that depends on ε > 0) overhead in the complexity. Instantiating this with the current best MPC algorithm for unweighted matching yields a (1 - ε)-approximation algorithm for maximum weighted matching that uses Oε(log log n) rounds, O(m/n) machines per round, and O(npoly(logn)) memory per machine. This improves upon the previous best approximation guarantee of (1/2-ε) for weighted graphs. In the context of single-pass streaming with random edge arrivals, our techniques yield a (1/2+c)-approximation algorithm thus breaking the natural barrier of 1/2.