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.

翻译：本文研究图上的机器人运动规划通用可解性（USolR）问题：给定无向图 G = (V, E) 和 p 个机器人，判定任意初始机器人的配置是否能够通过一系列有效且无碰撞的移动变换为任意目标配置。我们设计了一种规范累积过程，该过程将任意配置映射至占据固定顶点子集的配置，从而能够通过等价类分析配置的可达性。我们证明了在非通用可解的实例中，至少一半的配置从给定配置出发是不可达的，并利用这一性质设计了一种具有单侧错误的高效随机化算法；该算法可通过以 p 为倍数的运行时间开销进行去随机化。进一步，我们通过利用输入图 G = (V, E) 的结构优化了确定性算法，在稀疏图中实现了 O(p * (|V| + |E|)) 的运行时间，在稠密图中实现了 O(|V| + |E|) 的运行时间。最后，我们研究了通用可解性的图边增强（EAUS）问题：给定一个对 p 个机器人非通用可解的连通图 G，问题在于检查对于给定预算 b，是否可以通过添加至多 b 条边使 G 对 p 个机器人通用可解。我们给出了对于一般图 b 的上界为 p - 2。另一方面，我们也提供了需要添加 Theta(p) 条边的图实例。我们进一步研究了通用可解性的图顶点与边增强（VEAUS）问题，其中允许添加 a 个顶点和 b 条边，并给出了 a 和 b 的下界。