Pebble motion problems

Template:Short description 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 ("pebbles") 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 (in which the pebbles are robots) and network routing (in which the pebbles are packets of data). 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 GraphsTemplate:R/superscript formulated as follows:

Let $G=(V,E)$ be a graph with $n$ vertices. Let $P=\{1,\dots ,k\}$ be a set of pebbles with $k<n$. An arrangement of pebbles is a mapping $S:P\to V$ such that $S(i)\ne S(j)$ for $i\ne j$. A move $m=(p,u,v)$ consists of transferring pebble $p$ from vertex $u$ to adjacent unoccupied vertex $v$. The Pebble Motion on Graphs problem is to decide, given two arrangements $S_0$ and $S_{+}$, whether there is a sequence of moves that transforms $S_0$ into $S_{+}$.

Variations

Common variations on the problem limit the structure of the graph to be:

• a treeTemplate:R/superscript
• a square grid,Template:R/superscript
• a bi-connected graph.Template:R/superscript

Another set of variations consider the case in which someTemplate:R/superscript or allTemplate:R/superscript of the pebbles are unlabeled and interchangeable.

Other versions of the problem seek not only to prove reachability but to find a (potentially optimal) sequence of moves (i.e. a plan) which performs the transformation.

Complexity

Finding the shortest solution sequence in the pebble motion on graphs problem (with labeled pebbles) is known to be NP-hardTemplate:R/superscript and APX-hard.Template:R/superscript The unlabeled problem can be solved in polynomial time when using the cost metric mentioned above (minimizing the total number of moves to adjacent vertices), but is NP-hard for other natural cost metrics.Template:R/superscript