Space Complexity of Reachability in Simple Path Graphs

Overview & Synopsis

One of the central questions in space complexity is whether nondeterministic logspace computations ($\textsf{NL}$) can be simulated in unambiguous logspace ($\textsf{UL}$). A stronger notion, \emph{reach unambiguity} ($\textsf{RUL} \subseteq \textsf{UL} \cap \textsf{co}\textsf{UL}$), requires every configuration reachable from the start to have exactly one computation path (Buntrock et.al. 91). While directed graph reachability is $\textsf{NL}$-complete, planar graph reachability is in $\textsf{UL}$ and undirected graph reachability is in deterministic logspace ($\textsf{L}$). In this work, we study the space complexity of reachability problem for restricted directed graph families and show the following. - For reach unambiguous graphs (graphs with at most one path from start vertex to every vertex), Lange (Lange 97) showed that reachability is in $\textsf{RUL}$. As our main result, we show that for simple path graphs (introduced in Kannan et.al. 08), which contains reach unambiguous graphs), where each vertex has at most one simple path from the start, we show that the reachability problem lies in $\textsf{UL} \cap \textsf{co}\textsf{UL}$. - Our first result can also be equivalently stated as: the recognition problem for simple path is in $\textsf{UL} \cap \textsf{co}\textsf{UL}$ if and only if the reachability problem restricted to simple path graphs is also in $\textsf{UL} \cap \textsf{co}\textsf{UL}$. Inspired by this, we investigate the complexity of graph recognition versus graph reachability for graph classes. Observe that for any graph class, solving reachability (for the class) also solves the recognition problem for that class. We show that for reach unambiguous graphs, solving recognition is as hard as solving reachability (making both of them $\textsf{RUL}$-complete). (Abstract shortened to meet arXiv limits)