# A Two-Pass Lower Bound for Semi-Streaming Maximum Matching

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- Ω(logRS(n)/logn)), where RS(n) denotes the maximum number of induced matchings of size Θ(n) in any n-vertex graph, i.e., the largest density of a Ruzsa-Szemeredi graph. Currently, it is known that n^Ω(1/loglogn)≤ RS(n) ≤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^Ω(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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