Exact Algorithms for Coloring Graphs While Avoiding Monochromatic Cycles

  • Fabrice Talla Nobibon
  • Cor Hurkens
  • Roel Leus
  • Frits C. R. Spieksma
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6124)


We consider the problem of deciding whether a given directed graph can be vertex partitioned into two acyclic subgraphs. Applications of this problem include testing rationality of collective consumption behavior, a subject in micro-economics. We identify classes of directed graphs for which the problem is easy and prove that the existence of a constant factor approximation algorithm is unlikely for an optimization version which maximizes the number of vertices that can be colored using two colors while avoiding monochromatic cycles. We present three exact algorithms, namely an integer-programming algorithm based on cycle identification, a backtracking algorithm, and a branch-and-check algorithm. We compare these three algorithms both on real-life instances and on randomly generated graphs. We find that for the latter set of graphs, every algorithm solves instances of considerable size within few seconds; however, the CPU time of the integer-programming algorithm increases with the number of vertices in the graph while that of the two other procedures does not. For every algorithm, we also study empirically the transition from a high to a low probability of YES answer as function of a parameter of the problem. For real-life instances, the integer-programming algorithm fails to solve the largest instance after one hour while the other two algorithms solve it in about ten minutes.


Directed Graph Exact Algorithm Color Class Strongly Connect Component Dominance Rule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs (1993)Google Scholar
  2. 2.
    Apollonio, N., Franciosa, P.G.: A characterization of partial directed line graphs. Discrete Mathematics 307, 2598–2614 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chen, Z.: Efficient algorithm for acyclic colorings of graphs. Theoretical Computer Science 230, 75–95 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cherchye, L., De Rock, B., Vermeulen, F.: The collective model of household consumption: a nonparametric characterization. Econometrica 75, 553–574 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cherchye, L., De Rock, B., Sabbe, J., Vermeulen, F.: Nonparametric tests of collectively rational consumption behavior: an integer programming procedure. Journal of Econometrics 147, 258–265 (2008)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms. The MIT Press, Cambridge (2005)Google Scholar
  7. 7.
    Deb, R.: Acyclic partitioning problem is NP-complete for k = 2. Private communication. Yale University, United States (2008)Google Scholar
  8. 8.
    Deb, R.: An efficient nonparametric test of the collective household model. Working paper. Yale University, United States (2008)Google Scholar
  9. 9.
    Farin, G.: Class A Bézier curves. Computer Aided Geometric Design 23, 573–581 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hogg, T.: Refining the phase transition in combinatorial search. Artificial Intelligence 81, 127–154 (1985)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Monasson, R., Zecchina, R., Kirkpatrick, S., Selman, B., Troyansky, L.: Determining computational complexity from characteristic ‘phase transitions’. Nature 400, 133–137 (1999)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Talla Nobibon, F., Cherchye, L., De Rock, B., Sabbe, J., Spieksma, F.C.R.: Heuristics for deciding collectively rational consumption behavior. Research report ces08.24, University of Leuven (2008)Google Scholar
  13. 13.
    Talla Nobibon, F., Hurken, C., Leus, R., Spieksma, F.C.R.: Coloring graphs to avoid monochromatic cycles. University of Leuven (2009) (manuscript)Google Scholar
  14. 14.
    Tarjan, R.E.: Depth-first search and linear graph algorithms. SIAM Journal on Computing 2, 146–160 (1972)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Varian, H.: Revealed preference. In: Samuelsonian Economics and the 21st Century (2006)Google Scholar
  16. 16.
    Wu, Y., Yuan, J., Zhao, Y.: Partition a graph into two induced forests. Journal of Mathematical Study 1, 1–6 (1996)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Fabrice Talla Nobibon
    • 1
  • Cor Hurkens
    • 2
  • Roel Leus
    • 1
  • Frits C. R. Spieksma
    • 1
  1. 1.Operations Research GroupUniversity of LeuvenLeuvenBelgium
  2. 2.Department of Mathematics and Computer Science.Eindhoven University of TechnologyEindhoventhe Netherlands

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