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 n2βo(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(n7/6) space algorithms, leaving open a large gap between
(trivial) upper bounds and lower bounds