Abstract
We initiate a systematic study of the Row Deletion( B ) problem on matrices: For a fixed “forbidden submatrix” B, the question is, given an input matrix A (both A and B have entries chosen from a finite-size alphabet), to remove a minimum number of rows such that A has no submatrix which is equivalent to a row or column permutation of B. An application of this question can be found, e.g., in the construction of perfect phylogenies. Establishing a strong connection to variants of the NP-complete Hitting Set problem, we show that for most matrices B Row Deletion( B ) is NP-complete. On the positive side, the relation with Hitting Set problems yields constant-factor approximation algorithms and fixed-parameter tractability results.
Supported by the Deutsche Forschungsgemeinschaft (DFG), project PEAL (parameterized complexity and exact algorithms), NI 369/1; project OPAL (optimal solutions for hard problems in computational biology), NI 369/2; junior research group “PIAF” (fixed-parameter algorithms), NI 369/4.
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References
Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications (1999)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)
Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Graph-Modeled Data Clustering: Fixed-Parameter Algorithms for Clique Generation. In: Petreschi, R., Persiano, G., Silvestri, R. (eds.) CIAC 2003. LNCS, vol. 2653, pp. 108–119. Springer, Heidelberg (2003)
Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Automated Generation of Search Tree Algorithms for Graph Modification Problems. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 642–653. Springer, Heidelberg (2003)
Klinz, B., Rudolf, R., Woeginger, G.J.: Permuting Matrices to Avoid Forbidden Submatrices. Discrete Applied Mathematics 60, 223–248 (1995)
Natanzon, A., Shamir, R., Sharan, R.: Complexity Classification of Some Edge Modification Problems. Discrete Applied Mathematics 113, 109–128 (2001)
Niedermeier, R., Rossmanith, P.: An Efficient Fixed-Parameter Algorithm for 3- Hitting Set. Journal of Discrete Algorithms 1, 89–102 (2003)
Pe’er, I., Shamir, R., Sharan, R.: On the Generality of Phylogenies from Incomplete Directed Characters. In: Penttonen, M., Schmidt, E.M. (eds.) SWAT 2002. LNCS, vol. 2368, pp. 358–367. Springer, Heidelberg (2002)
Shamir, R., Sharan, R., Tsur, D.: Cluster Graph Modification Problems. In: Kučera, L. (ed.) WG 2002. LNCS, vol. 2573, pp. 379–390. Springer, Heidelberg (2002)
Sharan, R.: Graph Modification Problems and Their Applications to Genomic Research. Ph.D. Thesis, School of Computer Science, Tel-Aviv University (2002)
Wernicke, S.: On the Algorithmic Tractability of Single Nucleotide Polymorphism (SNP) Analysis and Related Problems. Diploma Thesis, WSI für Informatik, Universit ät Tübingen (September 2003)
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Wernicke, S., Alber, J., Gramm, J., Guo, J., Niedermeier, R. (2004). Avoiding Forbidden Submatrices by Row Deletions. In: Van Emde Boas, P., Pokorný, J., Bieliková, M., Štuller, J. (eds) SOFSEM 2004: Theory and Practice of Computer Science. SOFSEM 2004. Lecture Notes in Computer Science, vol 2932. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24618-3_30
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DOI: https://doi.org/10.1007/978-3-540-24618-3_30
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