An $O(n^{\epsilon})$ Space and Polynomial Time Algorithm for Reachability in Directed Layered Planar Graphs

TL;DR

This paper presents a polynomial-time algorithm for directed layered planar graph reachability that uses significantly less space than previous methods, advancing understanding of space complexity in graph algorithms.

Contribution

It introduces an $O(n^{ ext{ extonehalf}})$ space and polynomial time algorithm for reachability in directed layered planar graphs, improving upon prior space bounds.

Findings

Reachability in directed layered planar graphs can be decided in polynomial time with $O(n^ ext{ extonehalf})$ space.

The result advances the understanding of space complexity for graph reachability problems.

Progress is made towards resolving whether reachability is in the class SC.

Abstract

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 $O(n^\epsilon)$ space, for any $\epsilon > 0$. 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.