Abstract
We show that for any constant t ≥ 2, k -Independent Set and k -Dominating Set in t-track interval graphs are W[1]-hard. This settles an open question recently raised by Fellows, Hermelin, Rosamond, and Vialette. We also give an FPT algorithm for k -Clique in t-interval graphs, parameterized by both k and t, with running time max { t O(k), 2O(klogk) } ·poly(n), where n is the number of vertices in the graph. This slightly improves the previous FPT algorithm by Fellows, Hermelin, Rosamond, and Vialette. Finally, we use the W[1]-hardness of k -Independent Set in t-track interval graphs to obtain the first parameterized intractability result for a recent bioinformatics problem called Maximal Strip Recovery (MSR). We show that MSR-d is W[1]-hard for any constant d ≥ 4 when the parameter is either the total length of the strips, or the total number of adjacencies in the strips, or the number of strips in the optimal solution.
Supported in part by NSF grant DBI-0743670.
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References
Alcón, L., Cerioli, M.R., de Figueiredo, C.M.H., Gutierrez, M., Meidanis, J.: Tree loop graphs. Discrete Applied Mathematics 155, 686–694 (2007)
Bafna, V., Narayanan, B., Ravi, R.: Nonoverlapping local alignments (weighted independent sets of axis-parallel rectangles). Discrete Applied Mathematics 71, 41–53 (1996)
Bar-Yehuda, R., Halldórsson, M.M., Naor, J(S.), Shachnai, H., Shapira, I.: Scheduling split intervals. SIAM Journal on Computing 36, 1–15 (2006)
Bulteau, L., Fertin, G., Rusu, I.: Maximal strip recovery problem with gaps: hardness and approximation algorithms. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 710–719. Springer, Heidelberg (2009)
Butman, A., Hermelin, D., Lewenstein, M., Rawitz, D.: Optimization problems in multiple-interval graphs. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), pp. 268–277 (2007)
Chen, Z., Fu, B., Jiang, M., Zhu, B.: On recovering syntenic blocks from comparative maps. Journal of Combinatorial Optimization 18, 307–318 (2009)
Crochemore, M., Hermelin, D., Landau, G.M., Rawitz, D., Vialette, S.: Approximating the 2-interval pattern problem. Theoretical Computer Science 395, 283–297 (2008)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1998)
Fellows, M.R., Hermelin, D., Rosamond, F., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theoretical Computer Science 410, 53–61 (2009)
Gambette, P., Vialette, S.: On restrictions of balanced 2-interval graphs. In: Brandstädt, A., Kratsch, D., Müller, H. (eds.) WG 2007. LNCS, vol. 4769, pp. 55–65. Springer, Heidelberg (2007)
Griggs, J.R.: Extremal values of the interval number of a graph, II. Discrete Mathematics 28, 37–47 (1979)
Griggs, J.R., West, D.B.: Extremal values of the interval number of a graph. SIAM Journal on Algebraic and Discrete Methods 1, 1–7 (1980)
Gyárfás, A., West, D.B.: Multitrack interval graphs. Congressus Numerantium 109, 109–116 (1995)
Jiang, M.: Approximation algorithms for predicting RNA secondary structures with arbitrary pseudoknots. IEEE/ACM Transactions on Computational Biology and Bioinformatics, doi:10.1109/TCBB.2008.109 (to appear)
Jiang, M.: Inapproximability of maximal strip recovery. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 616–625. Springer, Heidelberg (2009)
Jiang, M.: Inapproximability of maximal strip recovery: II (Submitted)
Joseph, D., Meidanis, J., Tiwari, P.: Determining DNA sequence similarity using maximum independent set algorithms for interval graphs. In: Nurmi, O., Ukkonen, E. (eds.) SWAT 1992. LNCS, vol. 621, pp. 326–337. Springer, Heidelberg (1992)
Trotter Jr., W.T., Harary, F.: On double and multiple interval graphs. Journal of Graph Theory 3, 205–211 (1979)
Vialette, S.: On the computational complexity of 2-interval pattern matching problems. Theoretical Computer Science 312, 223–249 (2004)
Wang, L., Zhu, B.: On the tractability of maximal strip recovery. In: Proceedings of the 6th Annual Conference on Theory and Applications of Models of Computation (TAMC 2009), pp. 400–409 (2009)
West, D.B., Shmoys, D.B.: Recognizing graphs with fixed interval number is NP-complete. Discrete Applied Mathematics 8, 295–305 (1984)
Zheng, C., Zhu, Q., Sankoff, D.: Removing noise and ambiguities from comparative maps in rearrangement analysis. IEEE/ACM Transactions on Computational Biology and Bioinformatics 4, 515–522 (2007)
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Jiang, M. (2010). On the Parameterized Complexity of Some Optimization Problems Related to Multiple-Interval Graphs. In: Amir, A., Parida, L. (eds) Combinatorial Pattern Matching. CPM 2010. Lecture Notes in Computer Science, vol 6129. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13509-5_12
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DOI: https://doi.org/10.1007/978-3-642-13509-5_12
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