On the Reachability Problem on Monoid-Labelled Undirected Graphs
For complexity theorists, this work refines the boundary between deterministic and nondeterministic logspace for a family of graph problems parameterized by monoid structure, revealing new tractable cases and dichotomies.
The paper studies the labelled reachability problem on undirected graphs with monoid-labelled edges, characterizing when the problem is in L versus NL-hard. Key results include: for any monoid, the problem is in L when the accepting element is the identity; for commutative monoids, it is in L for all accepting subsets; for certain union-of-groups monoids, it is in L; and for monoids BA2 and U, a dichotomy between L and NL-completeness holds.
The labelled reachability problem for undirected graphs with edges labelled by elements of a monoid $M$ (more generally, groupoids or magmas) captures the classes $\sf{L}$ and $\sf{NL}$. Given a graph $G(V, E)$ labelled by $ϕ~\colon E \to M$, $s,t \in V$ and an accepting subset $F \subseteq M$, the problem asks to test whether there is a walk $P$ from $s$ to $t$ in $G$ where $ϕ(P) \in F$. Ramaswamy et al. (2019) studied the variant where the accepting element is part of the input for aperiodic monoids and groups. Motivated by the success in designing space-bounded algorithms for the undirected graph reachability problem, we study the labelled reachability problem when the accepting set is also fixed. This reveals finer complexity bounds and dichotomies for the problem based on the monoid and the accepting set. Previous results imply that the problem is in $\sf{L}$ for any finite accepting subset when $M$ is a group or belongs to $\sf{DA}$. We prove the following (for finite monoids): 1) For any monoid $M$, the problem is in $\sf{L}$ when the accepting element is the identity of $M$. If the accepting element is an idempotent, under suitable constraints, the problem is $\sf{NL}$-hard. 2) For any commutative monoid $M$, the problem is in $\sf{L}$ for all $F \subseteq M$. 3) For any $\mathcal{L}(\mathcal{R})$-commutative union-of-groups (UoG) monoid $M$, the problem is in $\sf{L}$ for all $F\subseteq M$. We show deterministic logspace algorithms for UoG monoids that are neither $\mathcal{L}$-commutative nor $\mathcal{R}$-commutative, under certain constraints. 4) For the monoids $\sf{BA_2}$ and $\sf{U}$, we show a dichotomy: for all $F \subseteq M$, the problem is either $\sf{NL}$-complete or in $\sf{L}$. Our results exploit the connection between Green's relations in the UoG monoids and the properties of the product graph (a graph introduced by Ramaswamy et al. (2019)).