Abstract
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 [IPL 1987] showed that this problem is NP-complete even on split graphs if q is part of input, but gave a n O(q) algorithm on chordal graphs. We first observe that the problem is W[2]-hard parameterized by q, even on split graphs. However, when parameterized by ℓ, the number of vertices in the solution, we give two fixed-parameter tractable algorithms.
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The first algorithm runs in time 5.44ℓ (n + #α(G))O(1) where #α(G) is the number of maximal independent sets of the input graph.
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The second algorithm runs in time q ℓ + o(ℓ) n O(1) T α where T α is the time required to find a maximum independent set in any induced subgraph of G.
The first algorithm is efficient when the input graph contains only polynomially many maximal independent sets; for example split graphs and co-chordal graphs. The running time of the second algorithm is FPT in ℓ alone (whenever T α is a polynomial in n), since q ≤ ℓ for all non-trivial situations. Finally, we show that (under standard complexity-theoretic assumptions) the problem does not admit a polynomial kernel on split and perfect graphs in the following sense:
(a) On split graphs, we do not expect a polynomial kernel if q is a part of the input.
(b) On perfect graphs, we do not expect a polynomial kernel even for fixed values of q ≥ 2.
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References
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. Discrete Applied Mathematics 158(7), 765–770 (2010)
Bodlaender, H.L.: Kernelization: New upper and lower bound techniques. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 17–37. Springer, Heidelberg (2009)
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)
Byskov, J.M.: Algorithms for k-colouring and finding maximal independent sets. In: SODA, pp. 456–457 (2003)
Byskov, J.M.: Enumerating maximal independent sets with applications to graph colouring. Oper. Res. Lett. 32(6), 547–556 (2004)
Dabrowski, K., Lozin, V.V., Müller, H., Rautenbach, D.: Parameterized algorithms for the independent set problem in some hereditary graph classes. In: Iliopoulos, C.S., Smyth, W.F. (eds.) IWOCA 2010. LNCS, vol. 6460, pp. 1–9. Springer, Heidelberg (2011)
Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In: STOC, pp. 251–260 (2010)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer-Verlag New York, Inc., Secaucus (2006)
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)
Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. In: STOC, pp. 133–142 (2008)
Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)
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)
Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: On generating all maximal independent sets. Inf. Process. Lett. 27(3), 119–123 (1988)
Khot, S., Raman, V.: Parameterized complexity of finding subgraphs with hereditary properties. Theor. Comput. Sci. 289(2), 997–1008 (2002)
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)
Naor, M., Schulman, L.J., Srinivasan, A.: Splitters and near-optimal derandomization. In: FOCS, pp. 182–191 (1995)
Nederlof, J.: Fast polynomial-space algorithms using möbius inversion: Improving on steiner tree and related problems. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 713–725. Springer, Heidelberg (2009)
Niedermeier, R.: Invitation to Fixed Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, USA (2006)
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)
Yannakakis, M., Gavril, F.: The maximum k-colorable subgraph problem for chordal graphs. Inf. Process. Lett. 24(2), 133–137 (1987)
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Misra, N., Panolan, F., Rai, A., Raman, V., Saurabh, S. (2013). Parameterized Algorithms for Max Colorable Induced Subgraph Problem on Perfect Graphs. In: Brandstädt, A., Jansen, K., Reischuk, R. (eds) Graph-Theoretic Concepts in Computer Science. WG 2013. Lecture Notes in Computer Science, vol 8165. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45043-3_32
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