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 $Θ(p)$ 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 $G = (V, E)$ 与 $p$ 个机器人，判定任意初始机器人的布局是否可通过一系列有效且无碰撞的移动变换为任意目标布局。我们设计了一种规范聚集过程，该过程可将任意布局映射至占据固定顶点子集的布局，从而通过等价类分析布局可达性。我们证明在非通用可解的实例中，至少半数布局从给定布局出发不可达，并利用此结论设计出具有单侧错误的高效随机算法；该算法可通过以 $p$ 倍运行时间为代价进行去随机化。进一步地，我们通过利用输入图 $G = (V, E)$ 的结构优化确定性算法，在稀疏图中实现 $\mathcal{O}(p \cdot (|V| + |E|))$ 的运行时间，在稠密图中实现 $\mathcal{O}(|V| + |E|)$ 的运行时间。最后，我们研究图边增强通用可解性（EAUS）问题：给定连通图 $G$ 对 $p$ 个机器人非通用可解，检查在给定预算 $b$ 下是否可通过添加至多 $b$ 条边使 $G$ 对 $p$ 个机器人通用可解。我们给出一般图中 $b$ 的上界为 $p - 2$，同时构造出需要添加 $Θ(p)$ 条边的图例。进一步研究图顶点与边增强通用可解性（VEAUS）问题（允许添加 $a$ 个顶点与 $b$ 条边），并给出 $a$ 与 $b$ 的下界。