Maximum Weight b-Matchings in Random-Order Streams

Abstract

We consider the maximum weight b-matching problem in the random-order semi-streaming model. Assuming all weights are small integers drawn from [1, W ], we present a 2 − 1 2W + ε approximation algorithm, using a memory of O(max (|MG|, n) • poly(log(m), W, 1/ε)), where |MG| denotes the cardinality of the optimal matching. Our result generalizes that of Bernstein [3], which achieves a 3/2 + ε approximation for the maximum cardinality simple matching. When W is small, our result also improves upon that of Gamlath et al. [11], which obtains a 2 − δ approximation (for some small constant δ ∼ 10 −17 ) for the maximum weight simple matching. In particular, for the weighted b-matching problem, ours is the first result beating the approximation ratio of 2. Our technique hinges on a generalized weighted version of edge-degree constrained subgraphs, originally developed by Bernstein and Stein [5]. Such a subgraph has bounded vertex degree (hence uses only a small number of edges), and can be easily computed. The fact that it contains a 2 − 1 2W + ε approximation of the maximum weight matching is proved using the classical Kőnig-Egerváry's duality theorem.