Universal Solvability for Robot Motion Planning on Graphs

TL;DR

This paper investigates the conditions under which robot motion planning on graphs is universally solvable, introduces algorithms for configuration reachability, and explores graph augmentation strategies to achieve universal solvability.

Contribution

It presents a canonical procedure for analyzing configuration reachability, efficient algorithms for determining solvability, and bounds on graph augmentation needed for universal solvability.

Findings

At least half of configurations are unreachable in non-universally solvable instances.

An efficient randomized algorithm with one-sided error is developed for USolR.

Upper bounds on edges needed for graph augmentation to achieve universal solvability are provided.

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 $\mathcal{O}(p \cdot (|V| + |E|))$ in sparse graphs and $\mathcal{O}(|V| + |E|)$ 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 $b$.