On the Reachability Problem on Monoid-Labelled Undirected Graphs

2026-06-19

AI Summary

This study addresses the problem of determining whether there exists a path between two vertices in an undirected graph whose edges are labeled by elements of a monoid, such that the product of the labels along the path belongs to a fixed accepting set. The work establishes, for the first time, a connection between Green’s relations in UoG monoids and structural properties of the associated product graphs. By integrating algebraic analysis, Green’s relation theory, and space-bounded computation models, it systematically characterizes how the algebraic properties of monoids influence the computational complexity of labeled reachability. Key contributions include proving that the problem lies in deterministic logspace (L) for several classes of monoids—including commutative and UoG monoids—revealing a dichotomy in complexity over BA₂ and U monoids, and precisely delineating boundaries between cases solvable in L and those that are NL-complete.

Abstract

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)).

Problem

• reachability
• monoid
• undirected graph
• accepting set
• computational complexity

Innovation

• monoid-labelled graphs
• logspace algorithms
• reachability dichotomy
• Green's relations
• union-of-groups monoids