This article describes scalability enhancements to a previously established belief propagation algorithm that solves bipartite maximum weight b-matching. The previous algorithm required O(|V | + |E|) space and O(|V ||E|) time, whereas we apply improvements to reduce the space to O(|V |) and thetimetoO(|V | 2.5) in the expected case (though worst case time is still O(|V ||E|)). The space improvement is most significant in cases where edge weights are determined by a function of node descriptors, such as a distance or kernel function. In practice, we demonstrate maximum weight b-matchings to be solvable on graphs with hundreds of millions of edges in only a few hours of compute time on a modern personal computer without parallelization, whereas neither the memory nor the time requirement of previously known algorithms would have allowed graphs of this scale. 1

Let G = (V, E) be a graph with positive integral edge weights. Our problem is to find a matching of maximum weight in G. We present a simple iterative algorithm for this problem that uses a maximum cardinality matching algorithm as a subroutine. Using the current fastest maximum cardinality matching algorithms, we solve the maximum weight matching problem in O(W √ nm logn(n 2 /m)) time, or in O(W n ω) time with high probability, where n = |V |, m = |E|, W is the largest edge weight, and ω < 2.376 is the exponent of matrix multiplication. In relatively dense graphs, our algorithm performs better than all existing algorithms with W = o(log 1.5 n). Our technique hinges on exploiting Edmonds ’ matching polytope and its dual. 1

In this paper we study linear-programming based approaches to the maximum matching problem in the semi-streaming model. In this model edges are presented sequentially, possibly in an adversarial order, and we are only allowed to use a small space. The allowed space is near linear in the number of vertices (and sublinear in the number of edges) of the input graph. The semi-streaming model is relevant in the context of processing of very large graphs. In recent years, there have been several new and exciting results in the semi-streaming model. However broad techniques such as linear programming have not been adapted to this model. In this paper we present several techniques to adapt and optimize linear-programming based approaches in the semi-streaming model. We use the maximum matching problem as a foil to demonstrate the e ectiveness of adapting such tools in this model. As a consequence we improve almost all previous results on the semi-streaming maximum matching problem. We also prove new results on interesting variants. 1