Universal Solvability for Robot Motion Planning on Graphs
Anubhav Dhar, Ashlesha Hota, Sudeshna Kolay, Pranav Nyati, Tanishq Prasad
TL;DR
This work formalizes Universal Solvability of Robot Motion Planning on Graphs (USolR), asking whether any two p-robot configurations on a graph G are mutually reachable under collision-free moves. It introduces a canonical accumulation procedure and proves that reachability partitions form equivalence classes of equal size, enabling linear-time randomized and polynomial-time deterministic algorithms to decide USolR, with optimized variants for dense or structured graphs. The authors also study augmentation problems EAUS and VEAUS, deriving upper and lower bounds on the number of edges or vertices needed to restore universal solvability, including tight results for 2-connected, 2-edge-connected, and 1-edge-connected graphs, and constructions that establish fundamental limits on augmentation. The results have practical relevance for reconfigurable multi-robot systems and constrained environments, offering tractable algorithms and principled graph-augmentation strategies to guarantee full reconfigurability. All methods are presented with rigorous complexity bounds and are applicable to a wide class of graph topologies via the accumulation framework and class-size symmetry.
Abstract
We study the Universal Solvability of Robot Motion Planning on Graphs (USolR) problem: given an undirected graph G = (V, E) and p robots, determine whether any arbitrary configuration of the robots can be transformed into any other arbitrary configuration via a sequence of valid, collision-free moves. We design a canonical accumulation procedure that maps arbitrary configurations to configurations that occupy a fixed subset of vertices, enabling us to analyze configuration reachability in terms of equivalence classes. We prove that in instances that are not universally solvable, at least half of all configurations are unreachable from a given one, and leverage this to design an efficient randomized algorithm with one-sided error, which can be derandomized with a blow-up in the running time by a factor of p. Further, we optimize our deterministic algorithm by using the structure of the input graph G = (V, E), achieving a running time of O(p * (|V| + |E|)) in sparse graphs and O(|V| + |E|) in dense graphs. Finally, we consider the Graph Edge Augmentation for Universal Solvability (EAUS) problem, where given a connected graph G that is not universally solvable for p robots, the question is to check if for a given budget b, at most b edges can be added to G to make it universally solvable for p robots. We provide an upper bound of p - 2 on b for general graphs. On the other hand, we also provide examples of graphs that require Theta(p) edges to be added. We further study the Graph Vertex and Edge Augmentation for Universal Solvability (VEAUS) problem, where a vertices and b edges can be added, and we provide lower bounds on a and b.