Reachability and distance computation are known
to be NL-complete in general graphs, but within
UL ? co-UL if the graphs are planar. However, finding
longest paths is known to be NP-complete, even
for planar graphs. We show that with the combination
of planarity and acyclicity, finding the length
of the longest path (and also enumerating one such
path) is also in UL ? co-UL. The result extends to
toroidal DAGs as well. We also address the question
of when reachability, distance, and longest path are
indeed equivalent on DAGs, and give partial bounds.
When the number of distinct paths is bounded by
a polynomial, counting the number of paths is known
to be in NL. We show that for planar DAGs with
this promise, counting can be done by a UAuxPDA
in polynomial time. The UAuxPDA(poly) bound also
holds if we want to compute the number of longest
paths, or shortest paths, and this number is bounded
by a polynomial (irrespective of the total number of
paths). Along the way, we show that counting in
general DAGs is possible in LogDCFL provided the
number of paths is bounded by a polynomial and the
target node is the only sink.