Feedback arc set problem and NP-hardness of minimum recurrent configuration problem of Chip-firing game on directed graphs

TL;DR

This paper investigates the computational complexity of finding minimum recurrent configurations in chip-firing games on Eulerian directed graphs, establishing NP-hardness and linking it to the feedback arc set problem.

Contribution

It demonstrates the NP-hardness of the minimum recurrent configuration problem and reveals its close relationship with the feedback arc set problem on Eulerian graphs.

Findings

MINREC is NP-hard on Eulerian directed graphs.

MINREC is closely related to the MINFAS problem.

Both problems are NP-hard.

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.