Near-Quadratic Lower Bounds for Two-Pass Graph Streaming Algorithms

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.