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 ("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 Graphs^[1] formulated as follows:

Let ${\displaystyle G=(V,E)}$ be a graph with ${\displaystyle n}$ vertices. Let ${\displaystyle P=\{1,\ldots ,k\}}$ be a set of pebbles with ${\displaystyle k<n}$. An arrangement of pebbles is a mapping ${\displaystyle S:P\rightarrow V}$ such that ${\displaystyle S(i)\neq S(j)}$ for ${\displaystyle i\neq j}$. A move ${\displaystyle m=(p,u,v)}$ consists of transferring pebble ${\displaystyle p}$ from vertex ${\displaystyle u}$ to adjacent unoccupied vertex ${\displaystyle v}$. The Pebble Motion on Graphs problem is to decide, given two arrangements ${\displaystyle S_{0}}$ and ${\displaystyle S_{+}}$, whether there is a sequence of moves that transforms ${\displaystyle S_{0}}$ into ${\displaystyle S_{+}}$.

Variations

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

• a tree^[2]
• a square grid,^[3]
• a bi-connected graph.^[4]

Another set of variations consider the case in which some^[5] or all^[3] 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-hard^[6] and APX-hard.^[3] 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.^[3]