A Two-Pass (Conditional) 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-Szemerédi graphs: Any two-pass semi-streaming algorithm for maximum matching has approximation ratio at most , 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-Szemerédi graph. Currently, it is known that 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) = nΩ(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.