Breaking the $n$-Pass Barrier: A Streaming Algorithm for Maximum Weight Bipartite Matching

Given a weighted bipartite graph with $n$ vertices and $m$ edges, the \emph{maximum weight bipartite matching} problem is to find a set of vertex-disjoint edges with the maximum weight. This classic problem has been extensively studied for over a century. In this paper, we present a new streaming algorithm for the maximum weight bipartite matching problem that uses $\widetilde{O}(n)$ space and $\widetilde{O}(\sqrt{m})$ passes, which breaks the $n$-pass barrier. All the previous streaming algorithms either require $\Omega(n \log n)$ passes or only find an approximate solution. Our streaming algorithm constructs a subgraph with $n$ edges of the input graph in $\widetilde{O}(\sqrt{m})$ passes, such that the subgraph admits the optimal matching with good probability. Our method combines various ideas from different fields, most notably the construction of \emph{space-efficient} interior point method (IPM), SDD system solvers, the isolation lemma, and LP duality. To the best of our knowledge, this is the first work that implements the SDD solvers and IPMs in the streaming model in $\widetilde{O}(n)$ spaces for graph matrices; previous IPM algorithms only focus on optimizing the running time, regardless of the space usage.