Space Complexity of the Directed Reachability Problem over Surface-Embedded Graphs

The graph reachability problem, the computational task of deciding whether there is a path between two given nodes in a graph, is the canonical problem for studying space-bounded computations. Three central open questions regarding the space complexity of the reachability problem over directed graphs are: (1) improving Savitch’s \(O(\log ^2 n)\) space bound, (2) designing a polynomial-time algorithm with \(O(n^{1-\epsilon })\) space bound, and (3) designing an unambiguous non-deterministic log-space (UL) algorithm. These are well-known open questions in complexity theory, and solving any one of them will be a major breakthrough. We discuss some of the recent progress reported on these questions for certain subclasses of surface-embedded directed graphs.

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