Target Pebbling in Trees

Graph pebbling is a game played on graphs with pebbles on their vertices. A pebbling move removes two pebbles from one vertex and places one pebble on an adjacent vertex. A configuration $C$ is a supply of pebbles at various vertices of a graph $G$, and a distribution $D$ is a demand of pebbles at various vertices of $G$. The $D$-pebbling number, $\pi(G, D)$, of a graph $G$ is defined to be the minimum number $m$ such that every configuration of $m$ pebbles can satisfy the demand $D$ via pebbling moves. The special case in which $t$ pebbles are demanded on vertex $v$ is denoted $D=v^t$, and the $t$-fold pebbling number, $\pi_{t}(G)$, equals $\max_{v\in G}\pi(G,v^t)$. It was conjectured by Alc\'on, Gutierrez, and Hurlbert that the pebbling numbers of chordal graphs forbidding the pyramid graph can be calculated in polynomial time. Trees, of course, are the most prominent of such graphs. In 1989, Chung determined $\pi_t(T)$ for all trees $T$. In this paper, we provide a polynomial-time algorithm to compute the pebbling numbers $\pi(T,D)$ for all distributions $D$ on any tree $T$, and characterize maximum-size configurations that do not satisfy $D$.