, Volume 73, Issue 2, pp 371–388 | Cite as

Algorithmic and Hardness Results for the Colorful Components Problems



In this paper we investigate the colorful components framework, motivated by applications emerging from comparative genomics. The general goal is to remove a collection of edges from an undirected vertex-colored graph \(G\) such that in the resulting graph \(G'\) all the connected components are colorful (i.e., any two vertices of the same color belong to different connected components). We want \(G'\) to optimize an objective function, the selection of this function being specific to each problem in the framework. We analyze three objective functions, and thus, three different problems, which are believed to be relevant for the biological applications: minimizing the number of singleton vertices, maximizing the number of edges in the transitive closure, and minimizing the number of connected components. Our main result is a polynomial-time algorithm for the first problem. This result disproves the conjecture of Zheng et al. that the problem is \( NP\)-hard (assuming \(P \ne NP\)). Then, we show that the second problem is \( APX\)-hard, thus proving and strengthening the conjecture of Zheng et al. that the problem is \( NP\)-hard. Finally, we show that the third problem does not admit polynomial-time approximation within a factor of \(|V|^{1/14 - \epsilon }\) for any \(\epsilon > 0\), assuming \(P \ne NP\) (or within a factor of \(|V|^{1/2 - \epsilon }\), assuming \(ZPP \ne NP\)).


Colorful components Graph coloring Exact polynomial-time algorithms Hardness of approximation 


  1. 1.
    Ausiello, G., Protasi, M., Marchetti-Spaccamela, A., Gambosi, G., Crescenzi, P., Kann, V.: Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties, 1st ed. Springer, New York (1999)Google Scholar
  2. 2.
    Avidor, A., Langberg, M.: The multi-multiway cut problem. Theoretical Computer Science 377(1–3), 35–42 (2007)Google Scholar
  3. 3.
    Bellare, M., Goldreich, O., Sudan, M.: Free bits, PCPs, and nonapproximability—towards tight results. SIAM J. Comput. 27(3), 804–915 (1998)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bruckner, S., Hüffner, F., Komusiewicz, C., Niedermeier, R.: Evaluation of ILP-based approaches for partitioning into colorful components. In: Bonifaci, V., Demetrescu, C., Marchetti-Spaccamela, A. (eds.) SEA, volume 7933 of Lecture Notes in Computer Science, pp. 176–187. Springer, Berlin (2013)Google Scholar
  5. 5.
    Bruckner, S., Hüffner, F., Komusiewicz, C., Niedermeier, R., Thiel, S., Uhlmann, J.: Partitioning into colorful components by minimum edge deletions. In: Kärkkäinen, J., Stoye, J., (eds.) CPM, volume 7354 of Lecture Notes in Computer Science, pp. 56–69. Springer, Berlin (2012)Google Scholar
  6. 6.
    Feige, U., Kilian, J.: Zero knowledge and the chromatic number. J. Comput. Syst. Sci. 57(2), 187–199 (1998)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    He, G., Liu, J., Zhao, C.: Approximation algorithms for some graph partitioning problems. J. Graph Algorithms Appl. 4(2), 1–11 (2000)Google Scholar
  8. 8.
    Mushegian, A.R.: Foundations of Comparative Genomics. Elsevier, Amsterdam (2010)Google Scholar
  9. 9.
    Paz, A., Moran, S.: Non deterministic polynomial optimization problems and their approximations. Theor. Comput. Sci. 15(3), 251–277 (1981)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Sankoff, D.: OMG! Orthologs for multiple genomes—competing formulations—(keynote talk). In: Chen, J., Wang, J., Zelikovsky, A. (eds.) ISBRA, volume 6674 of Lecture Notes in Computer Science, pp. 2–3. Springer, Berlin (2011)Google Scholar
  11. 11.
    Savard, O.T., Swenson, K.M.: A graph-theoretic approach for inparalog detection. BMC Bioinform. 13(S-19), S16 (2012)Google Scholar
  12. 12.
    Zheng, C., Swenson, K.M., Lyons, E., Sankoff, D.: OMG! Orthologs in multiple genomes—competing graph-theoretical formulations. In: Przytycka, T.M., Sagot, M.-F. (eds.) WABI, volume 6833 of Lecture Notes in Computer Science, pp. 364–375. Springer, Berlin (2011)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany
  2. 2.Faculty of InformaticsMasaryk UniversityBrnoCzech Republic

Personalised recommendations