Universal Solvability for Robot Motion Planning on Graphs

2025-06-23

AI Summary

This paper investigates universal solvability (USolR) for multi-robot motion planning on undirected graphs—i.e., deciding whether a collision-free path exists between arbitrary initial and goal configurations. For unsolvable graphs, we propose an equivalence-class analysis framework based on canonical cumulative processes, establishing for the first time that at least half of all configurations are unreachable in non-universally-solvable cases. We further formalize two graph augmentation problems: Edge-Augmented USolR (EAUS) and Vertex-Edge-Augmented USolR (VEAUS). For EAUS, we provide tight complexity bounds: a deterministic *O*(*p*(|*V*| + |*E*|)) algorithm for sparse graphs and *O*(|*V*| + |*E*|) for dense graphs; we prove a *p*−2 upper bound and an Ω(*p*) lower bound on the number of edges required, revealing a fundamental trade-off between augmentation budget and robot count.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Planning, Routing, and Scheduling: Replanning and Plan Repair
• Multiagent Systems: Multiagent Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• Determine if any robot configuration can reach any other on a graph
• Design efficient algorithms for universal solvability in sparse and dense graphs
• Study edge and vertex augmentation to achieve universal solvability

Innovation

• Canonical accumulation procedure for configuration mapping
• Efficient randomized algorithm with one-sided error
• Graph structure optimization for deterministic algorithm