On the Power of Unambiguity in Logspace

TL;DR

This paper advances understanding of the NL vs UL complexity classes by establishing new inclusions, exploring min-uniqueness, and introducing unambiguous variants of OptL, with implications for graph reachability problems.

Contribution

It proves that ReachFewL is contained in UL, explores the role of min-uniqueness in NL=UL, and introduces UOptL[log n] as an unambiguous class related to OptL[log n].

Findings

ReachFewL  UL unconditionally.

Min-uniqueness is necessary and sufficient for NL=UL.

Reachability on 3-page graphs is NL-complete.

Abstract

We report progress on the \NL vs \UL problem. [-] We show unconditionally that the complexity class $\ReachFewL\subseteq\UL$. This improves on the earlier known upper bound $\ReachFewL \subseteq \FewL$. [-] We investigate the complexity of min-uniqueness - a central notion in studying the \NL vs \UL problem. We show that min-uniqueness is necessary and sufficient for showing $\NL =\UL$. We revisit the class $\OptL[\log n]$ and show that {\sc ShortestPathLength} - computing the length of the shortest path in a DAG, is complete for $\OptL[\log n]$. We introduce $\UOptL[\log n]$, an unambiguous version of $\OptL[\log n]$, and show that (a) $\NL =\UL$ if and only if $\OptL[\log n] = \UOptL[\log n]$, (b) $\LogFew \leq \UOptL[\log n] \leq \SPL$. [-] We show that the reachability problem over graphs embedded on 3 pages is complete for \NL. This contrasts with the reachability problem over graphs embedded on 2 pages which is logspace equivalent to the reachability problem in planar graphs and hence is in \UL.