Pebble motion problems


The pebble motion problems, or pebble motion on graphs, are a set of related problems in graph theory dealing with the movement of multiple objects from vertex to vertex in a graph with a constraint on the number of pebbles that can occupy a vertex at any time. Pebble motion problems occur in domains such as multi-robot motion planning and network routing. The best-known example of a pebble motion problem is the famous 15 puzzle where a disordered group of fifteen tiles must be rearranged within a 4x4 grid by sliding one tile at a time.

Theoretical formulation

The general form of the pebble motion problem is Pebble Motion on Graphs formulated as follows:
Let be a graph with vertices. Let be a set of pebbles with. An arrangement of pebbles is a mapping such that for. A move consists of transferring pebble from vertex to adjacent unoccupied vertex. The Pebble Motion on Graphs problem is to decide, given two arrangements and, whether there is a sequence of moves that transforms into.

Variations

Common variations on the problem limit the structure of the graph to be:
Another set of variations consider the case in which some or all of the pebbles are unlabeled and interchangeable.
Other versions of the problem seek not only to prove reachability but to find a sequence of moves which performs the transformation.

Complexity

Finding the shortest path in the pebble motion on graphs problem is known to be NP-hard and APX-hard. The unlabeled problem can be solved in polynomial time when using the cost metric mentioned above, but is NP-hard for other natural cost metrics.