Fractional Matching Via Balls-and-Bins

In this paper we relate the problem of finding structures related to perfect matchings in bipartite graphs to a stochastic process similar to throwing balls into bins. Given a bipartite graph with n nodes on each side, we view each node on the left as having balls that it can throw into nodes on the right (bins) to which it is adjacent. If each node on the left throws exactly one ball and each bin on the right gets exactly one ball, then the edges represented by the ball-placement form a perfect matching. Further, if each thrower is allowed to throw a large but equal number of balls, and each bin on the right receives an equal number of balls, then the set of ball-placements corresponds to a perfect fractional matching – a weighted subgraph on all nodes with nonnegative weights on edges so that the total weight incident at each node is 1. We show that several simple algorithms based on throwing balls into bins deliver a near-perfect fractional matching. For example, we show that by iteratively picking a random node on the left and throwing a ball into its least-loaded neighbor, the distribution of balls obtained is no worse than randomly throwing kn balls into n bins. Another algorithm is based on the d-choice load-balancing of balls and bins. By picking a constant number of nodes on the left and appropriately inserting a ball into the least-loaded of their neighbors, we achieve a smoother load distribution on both sides – maximum load is at most log log n/ log d + O(1). When each vertex on the left throws k balls, we obtain an algorithm that achieves a load within k ± 1 on the right vertices.