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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