Abstract
The Convex Recoloring (CR) problem measures how far a tree of characters differs from exhibiting a so-called “perfect phylogeny”. For an input consisting of a vertex-colored tree T, the problem is to determine whether recoloring at most k vertices can achieve a convex coloring, meaning by this a coloring where each color class induces a subtree. The problem was introduced by Moran and Snir (J. Comput. Syst. Sci. 73:1078–1089, 2007; J. Comput. Syst. Sci. 74:850–869, 2008) who showed that CR is NP-hard, and described a search-tree based FPT algorithm with a running time of O(k(k/log k)k n 4). The Moran and Snir result did not provide any nontrivial kernelization. In this paper, we show that CR has a kernel of size O(k 2).
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Abu-Khzam, F.N., Collins, R.L., Fellows, M.R., Langston, M.A., Suters, W.H., Symons, C.T.: Kernelization algorithms for the vertex cover problem: Theory and experiments. In: Proc. 6th ACM-SIAM ALENEX, pp. 62–69. ACM-SIAM (2004)
Alber, J., Betzler, N., Niedermeier, R.: Experiments in data reduction for optimal domination in networks. Ann. Oper. Res. 146, 105–117 (2006)
Alber, J., Fellows, M.R., Niedermeier, R.: Polynomial-time data reduction for dominating sets. J. ACM 51, 363–384 (2004)
Bar-Yehuda, R., Feldman, I., Rawitz, D.: Improved approximation algorithm for convex recoloring of trees. Theory Comput. Syst. 43, 3–18 (2008)
Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 78, 160–174 (2009)
Bodlaender, H.L., Fellows, M.R., Langston, M., Ragan, M., Rosamond, F., Weyer, M.: Quadratic kernelization for convex recoloring of trees. In: Lin, G. (ed.) Proceedings 13th Annual International Computing and Combinatorics Conference, COCOON 2007, Heidelberg. Lecture Notes in Computer Science, vol. 4598, pp. 86–96. Springer, Berlin (2007)
Burrage, K., Estivill-Castro, V., Fellows, M.R., Langston, M.A., Mac, S., Rosamond, F.A.: The undirected feedback vertex set problem has a poly(k) kernel. In: Bodlaender, H.L., Langston, M.A. (eds.) Proceedings 2nd International Workshop on Parameterized and Exact Computation, IWPEC 2006. Lecture Notes in Computer Science, vol. 4169, pp. 192–202. Springer, Berlin (2006)
Chen, J., Fernau, H., Kanj, I.A., Xia, G.: Parametric duality and kernelization: Lower bounds and upper bounds on kernel size. SIAM J. Comput. 37, 1077–1106 (2007)
Chen, X., Hu, X., Shuai, T.: Inapproximability and approximability of maximal tree routing and coloring. J. Comb. Optim. 11, 219–229 (2006)
Courcelle, B.: The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Inf. Comput. 85, 12–75 (1990)
Courcelle, B.: Graph grammars, monadic second-order logic and the theory of graph minors. In: Robertson, N., Seymour, P. (eds.) Graph Structure Theory, Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference, Seattle WA, June 1991. Contemp. Math., vol. 147, pp. 565–590. Am. Math. Soc., Providence (1993)
Dom, M., Lokshtanov, D., Saurabh, S.: Incompressibility through colors and IDs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S.E., Thomas, W. (eds.) Automata, Languages and Programming, Proceedings of the 36th International Colloquium, ICALP 2009, Part I. Lecture Notes in Computer Science, vol. 5555, pp. 378–389. Springer, Berlin (2009)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006)
Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. In: Proceedings of the 38th Annual Symposium on Theory of Computing, STOC 2006, pp. 133–142. ACM, New York (2008)
Gramm, J., Nickelsen, A., Tantau, T.: Fixed-parameter algorithms in phylogenetics. Comput. J. 51, 79–101 (2008)
Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38, 31–45 (2007)
Hüffner, F.: Algorithms and experiments for parameterized approaches to hard graph problems. PhD thesis, Friedrich-Schiller University, Jena, Germany (2007)
Kammer, F., Tholey, T.: The complexity of minimum convex coloring. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) Proceedings 19th International Symposium on Algorithms and Computation, ISAAC 2008. Lecture Notes in Computer Science, vol. 5369, pp. 16–27. Springer, Berlin (2008)
Moran, S., Snir, S.: Efficient approximation of convex recolorings. J. Comput. Syst. Sci. 73, 1078–1089 (2007)
Moran, S., Snir, S.: Convex recolorings of strings and trees: Definitions, hardness results and algorithms. J. Comput. Syst. Sci. 74, 850–869 (2008)
Nemhauser, G.L., Trotter, L.E.: Vertex packing: Structural properties and algorithms. Math. Program. 8, 232–248 (1975)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, London (2006)
Ponta, O., Hüffner, F., Niedermeier, R.: Speeding up dynamic programming for some NP-hard graph recoloring problems. In: Agrawal, M., Du, D.-Z., Duan, Z., Li, A. (eds.) Proceedings 5th International Conference on Theory and Applications of Models of Computation, TAMC 2008. Lecture Notes in Computer Science, vol. 4978, pp. 490–501. Springer, Berlin (2008)
Thomassé, S.: A quadratic kernel for feedback vertex set. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, pp. 115–119 (2009)
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This research has been supported by the Australian Research Council Centre of Excellence in Bioinformatics, by the U.S. National Science Foundation under grant CCR–0311500, by the U.S. National Institutes of Health under grants 1-P01-DA-015027-01, 5-U01-AA-013512-02 and 1-R01-MH-074460-01, by the U.S. Department of Energy under the EPSCoR Laboratory Partnership Program and by the European Commission under the Sixth Framework Programme. The first author was partially supported by BSIK grant 03018 (BRICKS: Basic Research in Informatics for Creating the Knowledge Society). The second and fifth authors have been supported by a Fellowship to the Institute of Advanced Studies at Durham University, and hosted by a William Best Fellowship to Grey College during the preparation of the paper. A preliminary version of this paper was presented at COCOON 2007 [6].
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Bodlaender, H.L., Fellows, M.R., Langston, M.A. et al. Quadratic Kernelization for Convex Recoloring of Trees. Algorithmica 61, 362–388 (2011). https://doi.org/10.1007/s00453-010-9404-2
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DOI: https://doi.org/10.1007/s00453-010-9404-2