A Linear Time Algorithm for the Feasibility of Pebble Motion on Graphs

Given a connected, undirected, simple graph G = (V, E) and p ≤ |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|).