グラフ上のペブル移動問題の到達可能性を線形時間で判定するアルゴリズム

原文アブストラクト

Given a connected, undirected, simple graph $G = (V, E)$ and $p \le |V|$ pebbles labeled $1,..., p$, a configuration of these $p$ pebbles is an injective map assigning the pebbles to vertices of $G$. Let $S$ and $D$ be two such configurations. From a configuration, pebbles can move on $G$ as follows: In each step, at most one pebble may move from the vertex it currently occupies to an adjacent unoccupied vertex, yielding a new configuration. A natural question in this setting is the following: Is configuration $D$ reachable from $S$ and if so, how? We show that the feasibility of this problem can be decided in time $O(|V| + |E|)$.