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

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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.

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Authors and Affiliations

  1. Departamento de Ingeniería Matemática, Universidad de Chile, CMM (UMI 2807 - CNRS), Blanco Encalda, 2120, Santiago, Chile

    Kévin Perrot

  2. Aix Marseille Université, CNRS, LIF UMR 7279, 13288, Marseille, France

    Kévin Perrot

  3. Department of Mathematics for Computer Science, Institute of Mathematics, VAST, 18 Hoang Quoc Viet Road, Cau Giay District, Hanoi, Vietnam

    Trung Van Pham

Authors
  1. Kévin Perrot
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  2. Trung Van Pham
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Correspondence to Trung Van Pham.

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This paper was partially sponsored by Vietnam Institute for Advanced Study in Mathematics (VIASM), the Vietnamese National Foundation for Science and Technology Development (NAFOSTED), Fondecyt Postdoctoral Project 3140527, and Nucleo Milenio Informacion y Coordinacion en Redes (ACGO).

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Perrot, K., Van Pham, T. Feedback Arc Set Problem and NP-Hardness of Minimum Recurrent Configuration Problem of Chip-Firing Game on Directed Graphs. Ann. Comb. 19, 373–396 (2015). https://doi.org/10.1007/s00026-015-0266-9

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  • DOI: https://doi.org/10.1007/s00026-015-0266-9

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