, Volume 81, Issue 1, pp 26–46 | Cite as

Parameterized Algorithms for Max Colorable Induced Subgraph Problem on Perfect Graphs

  • Neeldhara Misra
  • Fahad Panolan
  • Ashutosh RaiEmail author
  • Venkatesh Raman
  • Saket Saurabh


We address the parameterized complexity of Max Colorable Induced Subgraph on perfect graphs. The problem asks for a maximum sized q-colorable induced subgraph of an input graph G. Yannakakis and Gavril (Inf Process Lett 24:133–137, 1987) showed that this problem is NP-complete even on split graphs if q is part of input, but gave an \(n^{O(q)}\) algorithm on chordal graphs. We first observe that the problem is W[2]-hard when parameterized by q, even on split graphs. However, when parameterized by \(\ell \), the number of vertices in the solution, we give two fixed-parameter tractable algorithms.
  • The first algorithm runs in time \(5.44^{\ell } (n+t)^{{\mathcal {O}}(1)}\) where t is the number of maximal independent sets of the input graph.

  • The second algorithm runs in time \({\mathcal {O}}(6.75^{\ell + o(\ell )} n^{{\mathcal {O}}(1)})\) on graph classes where the maximum independent set of an induced subgraph can be found in polynomial time.

The first algorithm is efficient when the input graph contains only polynomially many maximal independent sets; for example split graphs and co-chordal graphs. Finally, we show that (under standard complexity-theoretic assumption) the problem does not admit a polynomial kernel on split and perfect graphs in the following sense:
  1. (a)

    On split graphs, we do not expect a polynomial kernel if q is a part of the input.

  2. (b)

    On perfect graphs, we do not expect a polynomial kernel even for fixed values of \(q\ge 2\).



Maximum induced subgraphs Perfect graphs Co-chordal graphs Randomized FPT algorithms Polynomial kernel lower bounds 


  1. 1.
    Addario-Berry, L., Kennedy, W.S., King, A.D., Li, Z., Reed, B.A.: Finding a maximum-weight induced k-partite subgraph of an i-triangulated graph. Discret. Appl. Math. 158(7), 765–770 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alon, N., Gutin, G.Z., Kim, E.J., Szeider, S., Yeo, A.: Solving MAX-r-SAT above a tight lower bound. Algorithmica 61(3), 638–655 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Balas, E., Yu, C.S.: On graphs with polynomially solvable maximum-weight clique problem. Networks 19(2), 247–253 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier meets möbius: fast subset convolution. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pp. 67–74. San Diego, 11–13 June (2007)Google Scholar
  5. 5.
    Bodlaender, H.L., Thomassé, S., Yeo, A.: Analysis of data reduction: transformations give evidence for non-existence of polynomial kernels. In: Technical Report, (2008)Google Scholar
  6. 6.
    Bodlaender, H.L.: Kernelization: new upper and lower bound techniques. In: IWPEC, pp. 17–37. (2009)Google Scholar
  7. 7.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bodlaender, H.L., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: (Meta) kernelization. J. ACM 63(5), 44:1–44:69 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Byskov, J.M.: Algorithms for k-colouring and finding maximal independent sets. In: SODA, pp. 456–457. (2003)Google Scholar
  10. 10.
    Byskov, J.M.: Enumerating maximal independent sets with applications to graph colouring. Oper. Res. Lett. 32(6), 547–556 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dabrowski, K., Lozin, V.V., Müller, H., Rautenbach, D.: Parameterized algorithms for the independent set problem in some hereditary graph classes. In IWOCA, Volume 6460 of Lecture Notes in Computer Science, pp. 1–9. Springer, Berlin (2010)Google Scholar
  12. 12.
    Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM 61(4), 23:1–23:27 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dom, M., Lokshtanov, D., Saurabh, S.: Kernelization lower bounds through colors and ids. ACM Trans. Algorithms 11(2), 13:1–13:20 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  15. 15.
    Drange, P.G., Dregi, M.S., Fomin, F.V., Kreutzer, S., Lokshtanov, D., Pilipczuk, M., Pilipczuk, M. Reidl, F., Villaamil, F.S., Saurabh, S., Siebertz, S., Sikdar, S.: Kernelization and sparseness: the case of dominating set. In: 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016, pp. 31:1–31:14. Orléans, 17–20 February 2016Google Scholar
  16. 16.
    du Cray, H.P., Sau, I.: Improved FPT algorithms for weighted independent set in bull-free graphs. Discrete Math. 341(2), 451–462 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Flum, J., Grohe, M.: Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series). Springer, New York (2006)Google Scholar
  18. 18.
    Fomin, F.V., Gaspers, S., Pyatkin, A.V., Razgon, I.: On the minimum feedback vertex set problem: exact and enumeration algorithms. Algorithmica 52(2), 293–307 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fomin, F.V., Lokshtanov, D., Saurabh, S., Thilikos, D.M.: Bidimensionality and kernels. In: SODA, pp. 503–510, (2010)Google Scholar
  20. 20.
    Fomin, F.V., Lokshtanov, D., Panolan, F., Saurabh, S.: Efficient computation of representative families with applications in parameterized and exact algorithms. J. ACM 63(4), 29:1–29:60 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct pcps for NP. J. Comput. Syst. Sci. 77(1), 91–106 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ghosh, E., Kolay, S., Kumar, M., Misra, P., Panolan, F., Rai, A., Ramanujan, M.S.: Faster parameterized algorithms for deletion to split graphs. Algorithmica 71(4), 989–1006 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  24. 24.
    Gupta, S., Raman, V., Saurabh, S.: Maximum r-regular induced subgraph problem: fast exponential algorithms and combinatorial bounds. SIAM J. Discrete Math. 26(4), 1758–1780 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hardy, G.H., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proc. Lond. Math. Soc. 17, 75–115 (1918)CrossRefzbMATHGoogle Scholar
  26. 26.
    Hols, E.C., Kratsch, S.: A randomized polynomial kernel for subset feedback vertex set. Theor. Comput. Syst. 62(1), 63–92 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Khot, S., Raman, V.: Parameterized complexity of finding subgraphs with hereditary properties. Theor. Comput. Sci. 289(2), 997–1008 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kratsch, S., Wahlström, M.: Representative sets and irrelevant vertices: new tools for kernelization. In: 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, pp. 450–459. New Brunswick, 20–23 October 2012Google Scholar
  29. 29.
    Kratsch, S., Wahlström, M.: Compression via matroids: a randomized polynomial kernel for odd cycle transversal. ACM Trans. Algorithms 10(4), 20:1–20:15 (2014)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Lovasz, L.: Perfect graphs. In: Beineke, L.W., Wilson, R.J. (eds.) Selected Topics in Graph Theory, vol. 2, pp. 55–67. Academic Press, London (1983)Google Scholar
  31. 31.
    Naor, M., Schulman, L.J., Srinivasan, A.: Splitters and near-optimal derandomization. In FOCS, pp. 182–191. (1995)Google Scholar
  32. 32.
    Niedermeier, R.: Invitation to Fixed Parameter Algorithms (Oxford Lecture Series in Mathematics and Its Applications). Oxford University Press, Oxford (2006)CrossRefGoogle Scholar
  33. 33.
    Philip, G., Rai, A., Saurabh, S.: Generalized pseudoforest deletion: algorithms and uniform kernel. In: Mathematical Foundations of Computer Science 2015—40th International Symposium Part II, MFCS 2015, Milan, pp. 517–528. 24–28 August 2015Google Scholar
  34. 34.
    Raman, V., Saurabh, S., Sikdar, S.: Efficient exact algorithms through enumerating maximal independent sets and other techniques. Theor. Comput. Syst. 41(3), 563–587 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Raman, V., Saurabh, S.: Short cycles make w-hard problems hard: FPT algorithms for w-hard problems in graphs with no short cycles. Algorithmica 52(2), 203–225 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Thomassé, S., Trotignon, N., Vuskovic, K.: A polynomial turing-kernel for weighted independent set in bull-free graphs. Algorithmica 77(3), 619–641 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Trotignon, N.: Perfect graphs: a survey. CoRR, arXiv:abs/1301.5149, (2013)
  38. 38.
    Tsukiyama, S., Ide, M., Ariyoshi, H., Shirakawa, I.: A new algorithm for generating all the maximal independent sets. SIAM J. Comput. 6(3), 505–517 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Yannakakis, M., Gavril, F.: The maximum k-colorable subgraph problem for chordal graphs. Inf. Process. Lett. 24(2), 133–137 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Neeldhara Misra
    • 1
  • Fahad Panolan
    • 3
  • Ashutosh Rai
    • 2
    Email author
  • Venkatesh Raman
    • 2
  • Saket Saurabh
    • 2
    • 3
  1. 1.Indian Institute of TechnologyGandhinagarIndia
  2. 2.The Institute of Mathematical SciencesChennaiIndia
  3. 3.University of BergenBergenNorway

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