Abstract
Partitioning a permutation into a minimum number of monotone subsequences is \({\mathcal NP}\)-hard. We extend this complexity result to minimum partitioning into k-modal subsequences, that is, subsequences having at most k internal extrema. Based on a network flow interpretation we formulate both, the monotone and the k-modal version, as mixed integer programs. This is the first proposal to obtain provably optimal partitions of permutations. From these models we derive an LP rounding algorithm which is a 2-approximation for minimum monotone partitions and a (k+1)-approximation for minimum (upper) k-modal partitions in general; this is the first approximation algorithm for this problem. In computational experiments we see that the rounding algorithm performs even better in practice. For the associated online problem, in which the permutation becomes known to an algorithm sequentially, we derive a logarithmic lower bound on the competitive ratio for minimum monotone partitions, and we analyze two (bin packing) online algorithms. These findings immediately apply to online cocoloring of permutation graphs; they are the first results concerning online algorithms for this graph theoretical interpretation.
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References
Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms and Applications. Prentice-Hall, Inc., Englewood Cliffs (1993)
Bar-Yehuda, R., Fogel, S.: Partitioning a sequence into few monotone subsequences. Acta Inform. 35(5), 421–440 (1998)
Blasum, U., Bussieck, M.R., Hochstättler, W., Moll, C., Scheel, H.-H., Winter, T.: Scheduling trams in the morning. Math. Methods Oper. Res. 49(1), 137–148 (1999)
Brandstädt, A., Kratsch, D.: On partitions of permutations into increasing and decreasing subsequences. Elektron. Informationsverarb. Kybernet. 22(5/6), 263–273 (1986)
Chung, F.R.K.: On unimodal subsequences. J. Combin. Theory Ser. A 29, 267–279 (1980)
Di Stefano, G., Koči, M.L.: A graph theoretical approach to the shunting problem. Electr. Notes Theor. Comput. Sci. 92, 16–33 (2004)
Fiat, A., Woeginger, G.J.: Dagstuhl Seminar 1996. LNCS, vol. 1442. Springer, Heidelberg (1998)
Fomin, F.V., Kratsch, D., Novelli, J.-C.: Approximating minimum cocolourings. Inform. Process. Lett. 84(5), 285–290 (2002)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003)
Steele, J.M.: Long unimodal subsequences: A problem of F.R.K. Chung. Discrete Math. 33, 223–225 (1981)
Steele, J.M.: Variations on the monotone subsequence theme of Erdös and Szekeres. In: Aldous, D., Diaconis, P., Spencer, J., Steele, J.M. (eds.) Discrete Probability and Algorithms, pp. 111–131. Springer, New York (1995)
Wagner, K.: Monotonic coverings of finite sets. Elektron. Informationsverarb. Kybernet. 20(12), 633–639 (1984)
Winter, T., Zimmermann, U.T.: Real-time dispatch of trams in storage yards. Ann. Oper. Res. 96, 287–315 (2000)
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Di Stefano, G., Krause, S., Lübbecke, M.E., Zimmermann, U.T. (2006). On Minimum k-Modal Partitions of Permutations. In: Correa, J.R., Hevia, A., Kiwi, M. (eds) LATIN 2006: Theoretical Informatics. LATIN 2006. Lecture Notes in Computer Science, vol 3887. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11682462_36
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DOI: https://doi.org/10.1007/11682462_36
Publisher Name: Springer, Berlin, Heidelberg
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