Feedback Arc Set Problem and NP-Hardness of Minimum Recurrent Configuration Problem of Chip-Firing Game on Directed Graphs

Abstract

In this paper we present further studies of recurrent configurations of chip-firing games on Eulerian directed graphs (simple digraphs), a class on the way from undirected graphs to general directed graphs. A computational problem that arises naturally from this model is to find the minimum number of chips of a recurrent configuration, which we call the minimum recurrent configuration (MINREC) problem. We point out a close relationship between MINREC and the minimum feedback arc set (MINFAS) problem on Eulerian directed graphs, and prove that both problems are NP-hard.