The halting problem for chip-firing on finite directed graphs

We consider a chip-firing game on finite directed graphs and give an answer to a question posed by Bjorner, Lovasz, and Shor in 1991: given an initial configuration of chips, when does it stabilize? The approach they took to address this halting problem involves computing a period vector p with the property that toppling the vertices according to p results in the original configuration, and then checking if it is possible to topple according to p legally (without any of the vertices ever having negative chips). This approach is problematic because the entries of p can grow exponentially in the number of vertices. We make precise a measure of “Eulerianness” and show that, in addition to graphs with a high degree of Eulerianness, for relatively “anti-Eulerian” graphs you can do much better. In addition, we take steps toward a potential proof that the problem is NP-Hard by reducing an NP-Hard problem in the context of another chip-firing game, called the dollar game, to our problem (where the number of edges making up the graphs could be a very fastgrowing function of the number of vertices). The ideas developed in the course of answering this stabilization question give rise to a natural generalization of the BEST theorem to general directed graphs.