We provide algorithms with improved pass and space complexities for approximately solving linear programs, optimal transport, bipartite matching, and more in the semi-streaming model. For instance, we provide a (randomized)algorithm computing a maximum cardinality matching in an unweighted bipartite graph in O(\log^2 n \cdot \epsilon ^{-1} ) passes, using O(n \log^2 n \cdot \epsilon^{-1)auxiliary memory. This marks the first improvements to the O(\log\log\epsilon^{-1}\cdot \epsilon^{-2})pass, O(n \log\log\epsilon}   ^{-1} \cdot \epsilon^{-2 ) -space algorithms of AG13%DIFDELCMD < ] %%% when \epsilon is moderately small . To obtain our results , we give an O(n) space deterministic semi-streaming algorithm for approximating the value of linear programs (in the form of box-simplex games), based on low-space implementations of She17 , JST19 . We further give a general sampling procedure for explicitly forming a fractional solution in low space, yielding improved semi-streaming guarantees for optimal transport and, in some regimes, maximum weighted matching. Finally, we improve the space complexity of our maximum cardinality matching method using an implicit implementation of the random walk rounding of%DIFDELCMD < [%%% GKK10 via custom turnstile samplers .
We provide (\epsilon ^{-1} )-pass semi-streaming algorithms for computing (1-\epsilon)-approximate maximum cardinality matchings in bipartite graphs. Our most efficient methods are deterministic and use optimal, O(n )auxiliary memory. This marks the first improvements to the O(\log\log\epsilon^{-1}\cdot \epsilon^{-2})pass, O(n \log\log\epsilon} ), space, improving upon the space complexity of the previous state-of-the-art (\epsilon ^{-1} ) %DIFDELCMD < ] %%% -pass algorithm of Ahn and Gupta . To obtain our results we provide semi-streaming adaptations of more general continuous optimization tools. Further, we leverage these techniques to obtain improvements for streaming variants of approximate linear programming, optimal transport , %DIFDELCMD < [%%% and exact matching .
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algorithms with improved pass and space complexities for approximately solving linear programs, optimal transport, bipartite matching, and more in the semi-streaming model. For instance, we provide a (randomized)algorithm computing a maximum cardinality matching in an unweighted bipartite graph in O(\log^2 n \cdot \epsilon
(\epsilon
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) passes, using
)-pass semi-streaming algorithms for computing (1-\epsilon)-approximate maximum cardinality matchings in bipartite graphs. Our most efficient methods are deterministic and use optimal,
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\log^2 n \cdot \epsilon^{-1
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), space, improving upon the space complexity of the previous state-of-the-art
A
(\epsilon
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\cdot \epsilon^{-2
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-space algorithms of
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AG13
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when \epsilon is moderately small
-pass algorithm of Ahn and Gupta
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, we give an O(n) space deterministic
we provide
D
algorithm for approximating the value of linear programs (in the form of box-simplex games), based on low-space implementations of
R
She17
adaptations of more general continuous optimization tools. Further, we leverage these techniques to obtain improvements for streaming variants of approximate linear programming, optimal transport
D
JST19
D
. We further give a general sampling procedure for explicitly forming a fractional solution in low space, yielding improved semi-streaming guarantees for optimal transport and, in some regimes, maximum weighted matching. Finally, we improve the space complexity of our maximum cardinality matching method using an implicit implementation of the random walk rounding of
D
GKK10
R
via custom turnstile samplers
and exact matching