Computing sandpile configurations using integer linear programming

Abstract

Sandpile group or Abelian sandpile model was the first example of a self-organized critical system studied by Bak, Tang and Wiesenfeld. It is well known that these recurrent configurations can be characterized as the optimal solution of certain non-linear optimization problems. We show that recurrent configurations of the Abelian sandpile model of a graph correspond to optimal solutions of some integer linear programs. More precisely, we present two new integer linear programming models, one that computes recurrent configurations and other one that computes the order of the configuration. As an application, we calculate the identity configuration for the cone of a regular graph and the cycle with n+1 vertices using the Weak Duality Theorem in linear programming.