An $O(n^ε)$ Space and Polynomial Time Algorithm for Reachability in Directed Layered Planar Graphs

Given a graph $G$ and two vertices $s$ and $t$ in it, {\em graph reachability} is the problem of checking whether there exists a path from $s$ to $t$ in $G$. We show that reachability in directed layered planar graphs can be decided in polynomial time and $O(n^ε)$ space, for any $ε> 0$. The previous best known space bound for this problem with polynomial time was approximately $O(\sqrt{n})$ space \cite{INPVW13}. Deciding graph reachability in {\SC} is an important open question in complexity theory and in this paper we make progress towards resolving this question.