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)/log(n))),
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.