Directed Planar Reachability Is in Unambiguous Log-Space

Abstract

We make progress in understanding the complexity of the graph reachability problem in the context of unambiguous logarithmic space computation; a restricted form of nondeterminism. As our main result, we show a new upper bound on the directed planar reachability problem by showing that it can be decided in the class unambiguous logarithmic space (UL). We explore the possibility of showing the same upper bound for the general graph reachability problem. We give a simple reduction showing that the reachability problem for directed graphs with thickness two is complete for the class nondeterministic logarithmic space (NL). Hence an extension of our results to directed graphs with thickness two will unconditionally collapse NL to UL.