Optimization Letters

, Volume 12, Issue 3, pp 443–454 | Cite as

On the complexity of rainbow spanning forest problem

  • Francesco Carrabs
  • Carmine Cerrone
  • Raffaele Cerulli
  • Selene Silvestri
Original Paper
  • 64 Downloads

Abstract

Given a graph \(G=(V,E,L)\) and a coloring function \(\ell : E \rightarrow L\), that assigns a color to each edge of G from a finite color set L, the rainbow spanning forest problem (RSFP) consists of finding a rainbow spanning forest of G such that the number of components is minimum. A spanning forest is rainbow if all its components (trees) are rainbow. A component whose edges have all different colors is called rainbow component. The RSFP on general graphs is known to be NP-complete. In this paper we use the 3-SAT Problem to prove that the RSFP is NP-complete on trees and we prove that the problem is solvable in polynomial time on paths, cycles and if the optimal solution value is equal to 1. Moreover, we provide an approximation algorithm for the RSFP on trees and we show that it approximates the optimal solution within 2.

Keywords

Graph theory Edge-colored graph Rainbow components Approximation algorithm 

References

  1. 1.
    Broersma, H., Li, X.: Spanning trees with many or few colors in edge-colored graphs. Graph Theory 17, 259–269 (1997)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brualdi, R.A., Hollingsworth, S.: Multicolored forests in complete bipartite graphs. Discret. Math. 240, 239–245 (2001)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Carr, R.D., Doddi, S., Konjedov, G., Marathe, M.: On the red-blue set cover problem. In: 11th ACN-SIAM Symposium on Discrete Algorithms, pp. 345–353 (2000)Google Scholar
  4. 4.
    Carrabs, F., Cerrone, C., Cerulli, R., Silvestri, S.: The rainbow spanning forest problem. Soft. Comput. pp. 1–12 (2017)Google Scholar
  5. 5.
    Carrabs, F., Cerrone, C., Cerulli, R.: A tabu search approach for the circle packing problem. In: IEEE 2014 17th International Conference on Network-Based Information Systems, pp. 165–171 (2014)Google Scholar
  6. 6.
    Carrabs, F., Cerulli, R., Dell’Olmo, P.: A mathematical programming approach for the maximum labeled clique problem. Proc. Soc. Behav. Sci. 108, 69–78 (2014)CrossRefGoogle Scholar
  7. 7.
    Carrabs, F., Cerulli, R., Gentili, M.: The labeled maximum matching problem. Comput. Oper. Res. 36, 1859–1871 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cerrone, C., Cerulli, R., Golden, B.: Carousel greedy: a generalized greedy algorithm with applications in optimization. Comput. Oper. Res. 85, 97–112 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cerrone, C., Cerulli, R., Gaudioso, M.: Omega one multi ethnic genetic approach. Optim. Lett. 10(2), 309–324 (2016)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cerrone, C., Cerulli, R., Gentili, M.: Vehicle-id sensor location for route flow recognition: Models and algorithms. Eur. J. Oper. Res. 247(2), 618–629 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cerulli, R., Fink, A., Gentili, M., Voß, S.: Extensions of the minimum labelling spanning tree problem. J. Telecommun. Inf. Technol. 4, 39–45 (2006)Google Scholar
  12. 12.
    Chang, R.S., Leu, S.J.: The minimum labeling spanning trees. Inf. Process. Lett. 63, 277–282 (1997)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Chen, Y., Cornick, N., Hall, A.O., Shajpal, R., Silberholz, J., Yahav, I., Golden, B.: Comparison of heuristics for solving the gmlst problem. In: Raghavan, S., Golden, B., Wasil, E. (eds.) Telecommunications Modeling, Policy, and Technology, pp. 191–217. Springer, Berlin (2008)CrossRefGoogle Scholar
  14. 14.
    Consoli, S., Darby-Dowman, K., Mladenović, N., Moreno-Pérez, J.A.: Variable neighbourhood search for the minimum labelling steiner tree problem. Ann. Oper. Res. 172, 71–96 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Consoli, S., Moreno-Pérez, J.A., Darby-Dowman, K., Mladenović, N.: Discrete particle swarm optimization for the minimum labelling steiner tree problem. Nat. Comput. 9, 29–46 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Jozefowiez, N., Laporte, G., Semet, F.: A branch-and-cut algorithm for the minimum labeling hamiltonian cycle problem and two variants. Comput. Oper. Res. 38, 1534–1542 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kapoor, S., Ramesh, H.: Algorithms for enumerating all spanning trees of undirected and weighted graphs. SIAM J. Comput. 24(2), 247–265 (1995)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Krumke, S., Wirth, H.: On the minimum label spanning tree problem. Inf. Process. Lett. 66(2), 81–85 (1998)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Li, X., Zhang, X.Y.: On the minimum monochromatic or multicolored subgraph partition problems. Theor. Comput. Sci. 385, 1–10 (2007)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Silvestri, S., Laporte, G., Cerulli, R.: The rainbow cycle cover problem. Networks 68, 260–270 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Suzuki, K.: A necessary and sufficient condition for the existence of a heterochromatic spanning tree in a graph. Graph Comb. 22, 261–269 (2006)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Xiong, Y., Golden, B., Wasil, E.: The colorful traveling salesman problem. In: Baker, E.K., Joseph, A., Mehrotra, A., Trick, M.A. (eds.) Extending the Horizons: Advances in Computing, Optimization, and Decision Technologies, pp. 115–123. Springer, Berlin (2007)CrossRefGoogle Scholar
  23. 23.
    Xiongm, Y., Golden, B., Wasil, E., Chen, S.: The label-constrained minimum spanning tree problem. In: Raghavan, S., Golden, B., Wasil, E. (eds.) Telecommunications Modeling, Policy, and Technology, pp. 39–58. Springer, Berlin (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SalernoFiscianoItaly
  2. 2.Department of Computer ScienceUniversity of SalernoFiscianoItaly
  3. 3.Department of Biosciences and TerritoryUniversity of MolisePescheItaly

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