On the order of the shortest solution sequences for the pebble motion problems

TL;DR

This paper addresses the Pebble Motion Problem on graphs by deriving tight upper bounds on the length of the shortest solution sequences and providing efficient algorithms. It introduces key structural notions such as isthmuses and centroid decompositions, and develops a main algorithm that reduces general instances to solvable subproblems on small subtrees, with linear-time components and subpolynomial factors. For trees, it achieves a move-length bound of $\mathcal{O}\big(n\mathrm{D}(G) + \min\{k n \mathrm{D}(G), \; n^{2} \log(1+\min\{n,k\})\}\big)$ and a near-linear running time, improving prior results; for general graphs, it extends the approach to obtain $\mathcal{O}\big(n\mathrm{D}(G) + \frac{n^{2}\min\{n,\mathrm{CL}(G)\}}{N-n} + n^{2}\log(1+\min\{n,N-n\})\big)$ with refinements yielding further improvements when $N-n$ is small. The refinements and corollaries (including when $N-n$ is constant) show that the problem becomes substantially more tractable than the classical $\mathcal{O}(N^{3})$ barrier in many practical regimes. Overall, the work provides a robust framework for motion planning in the PMP/MAPF family on both trees and general graphs, with implications for reconfiguration solvers in robotics and AI planning.

Abstract

Let $G$ be a connected graph with $N$ vertices. Let $k$ be the number of vertices in a longest path of $G$ such that every vertex on the path is a cut vertex of $G$, and every intermediate vertex of the path is a degree-two vertex of $G$. We conventionally set $k = 1$ when $G$ is $2$-edge-connected. Let $P=\{1,\ldots,n\}$ be a set of pebbles with $k < N-n$. A \textit{configuration} of $P$ on $G$ is defined as a function $f$ from $V(G)$ to $\{0, 1, \ldots, n \}$ with $|f^{-1}(i)| = 1$ for $1 \le i \le n$, where $f^{-1}(i)$ is a vertex occupied with the $i$th pebble for $1 \le i \le n$ and $f^{-1}(0)$ is a set of unoccupied vertices. A \textit{move} is defined as shifting a pebble from a vertex to some unoccupied neighbor. The {\it pebble motion problem on the pair $(G,P)$} is to decide whether a given configuration of pebbles is reachable from another by executing a sequence of moves. Let $\D(G)$ denote the diameter of the graph $G$, and let $\CL(G)$ denote the maximum length of a shortest cycle containing a vertex $v$, taken over all vertices $v$ in all $2$-connected components of $G$. For completeness, we define $\CL(G) := 1$ when $G$ is a tree. In this paper, we show that the length of the shortest solution sequences for the pebble motion problem on a pair $(G, P)$ is in $\Ord\left(n\D(G) + \min\left\{k n \D(G),\ n^{2} \log\big(1+\min\{n, k\}\big)\right\}\right)$ if $G$ is an $N$-vertex tree, and in $\Ord\left(n\D(G)+\frac{n^2\min\{n,\CL(G)\}}{N-n}+n^2\log(1+\min\{n, N-n\})\right)$ if $G$ is a connected general $N$-vertex graph. Furthermore, in the case where $G$ is a connected general $N$-vertex graph and the number of unoccupied spaces $N - n$ is bounded by some constant, this length admits an upper bound of $\Ord(n \CL(G) \D(G))$. Keywords: pebble motion, motion planning, multi-agent path finding, $15$-puzzle, tree

Theorems & Definitions (23)

• Theorem 1
• Theorem 2
• Conjecture 1
• Theorem 3
• Theorem 4
• Corollary 5
• Lemma 6
• Theorem 7
• Lemma 8
• Lemma 9