Breaking the n-Pass Barrier: A Streaming Algorithm for Maximum Weight Bipartite Matching
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Given a weighted bipartite graph with n vertices and m edges, the 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 O(n) space and O(√(m)) passes, which breaks the n-pass barrier. All the previous algorithms either require Ω(n log n) passes or only find an approximate solution. To achieve this pass bound, our algorithm combines a number of techniques from different fields such as the interior point method (IPM), symmetric diagonally dominant (SDD) system solving, the isolation lemma, and LP duality. To the best of our knowledge, this is the first work that implements the SDD solver and IPM in the streaming model in O(n) spaces for graph matrix. All the previous IPMs only focus on optimizing the running time, regardless of the space usage. The LP solver for general matrix is impossible to be implemented in O(n) spaces.
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