The Complexity of Graph Pebbling

In a graph $G$ whose vertices contain pebbles, a pebbling move $uv$ removes two pebbles from $u$ and adds one pebble to a neighbor $v$ of $u$. The optimal pebbling number ${\widehat{\pi}}(G)$ is the minimum $k$ such that there exists a distribution of $k$ pebbles to $G$ so that for any target vertex $r$ in $G$, there is a sequence of pebbling moves which places a pebble on $r$. The pebbling number $\pi(G)$ is the minimum $k$ such that for all distributions of $k$ pebbles to $G$ and for any target vertex $r$, there is a sequence of pebbling moves which places a pebble on $r$. We explore the computational complexity of computing ${\widehat{\pi}}(G)$ and $\pi(G)$. In particular, we show that deciding whether ${\widehat{\pi}}(G)\leq k$ is NP-complete. Furthermore, we prove that deciding whether $\pi(G)\leq k$ is ${\Pi_2^{\mathrm{P}}}$-complete and therefore both NP-hard and coNP-hard. Additionally, we provide a characterization of when an unordered set of pebbling moves can be ordered to form a valid sequence of pebbling moves.