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Register a new userA Two-Pass Lower Bound for Semi-Streaming Maximum Matching
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Sepehr Assadi
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Authors
Sepehr Assadi
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We prove a lower bound on the space complexity of two-pass semi-streaming algorithms that approximate the maximum matching problem. The lower bound is parameterized by the density of Ruzsa-Szemeredi graphs: * Any two-pass semi-streaming algorithm for maximum matching has approximation ratio at least $(1- Omega(frac{log{RS(n)}}{log{n}}))$, where $RS(n)$ denotes the maximum number of induced matchings of size $Theta(n)$ in any $n$-vertex graph, i.e., the largest density of a Ruzsa-Szemeredi graph. Currently, it is known that $n^{Omega(1/!loglog{n})} leq RS(n) leq frac{n}{2^{O(log^*{!(n)})}}$ and closing this (large) gap between upper and lower bounds has remained a notoriously difficult problem in combinatorics. Under the plausible hypothesis that $RS(n) = n^{Omega(1)}$, our lower bound is the first to rule out small-constant approximation two-pass semi-streaming algorithms for the maximum matching problem, making progress on a longstanding open question in the graph streaming literature.
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We prove that any two-pass graph streaming algorithm for the $s$-$t$ reachability problem in $n$-vertex directed graphs requires near-quadratic space of $n^{2-o(1)}$ bits. As a corollary, we also obtain near-quadratic space lower bounds for several other fundamental problems including maximum bipartite matching and (approximate) shortest path in undirected graphs. Our results collectively imply that a wide range of graph problems admit essentially no non-trivial streaming algorithm even when two passes over the input is allowed. Prior to our work, such impossibility results were only known for single-pass streaming algorithms, and the best two-pass lower bounds only ruled out $o(n^{7/6})$ space algorithms, leaving open a large gap between (trivial) upper bounds and lower bounds.
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