The Complexity of Graph Pebbling

Abstract

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.