Abstract
We extend previous studies on NP-hard problems dealing with the decomposition of nonnegative integer vectors into sums of few homogeneous segments. These problems are motivated by radiation therapy and database applications. If the segments may have only positive integer entries, then the problem is called Vector Explanation + . If arbitrary integer entries are allowed in the decomposition, then the problem is called Vector Explanation. Considering several natural parameterizations (including maximum vector entry, maximum difference between consecutive vector entries, maximum segment length), we obtain a refined picture of the computational (in-)tractability of these problems. In particular, we show that in relevant cases Vector Explanation + is algorithmically harder than Vector Explanation .
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Agarwal, D., Barman, D., Gunopulos, D., Young, N., Korn, F., Srivastava, D.: Efficient and effective explanation of change in hierarchical summaries. In: Proc. 13th KDD, pp. 6–15. ACM (2007)
de Azevedo Pribitkin, W.: Simple upper bounds for partition functions. The Ramanujan Journal 18, 113–119 (2009)
Baatar, D., Hamacher, H.W., Ehrgott, M., Woeginger, G.J.: Decomposition of integer matrices and multileaf collimator sequencing. Discrete Appl. Math. 152(1-3), 6–34 (2005)
Bansal, N., Chen, D.Z., Coppersmith, D., Hu, X.S., Luan, S., Misiolek, E., Schieber, B., Wang, C.: Shape rectangularization problems in intensity-modulated radiation therapy. Algorithmica 60(2), 421–450 (2011)
Biedl, T.C., Durocher, S., Hoos, H.H., Luan, S., Saia, J., Young, M.: A note on improving the performance of approximation algorithms for radiation therapy. Inf. Process. Lett. 111(7), 326–333 (2011)
Biedl, T.C., Durocher, S., Engelbeen, C., Fiorini, S., Young, M.: Faster optimal algorithms for segment minimization with small maximal value. Discrete Appl. Math. 161(3), 317–329 (2013)
Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Cross-composition: A new technique for kernelization lower bounds. In: Proc. 28th STACS. LIPIcs, vol. 9, pp. 165–176. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2011)
Cambazard, H., O’Mahony, E., O’Sullivan, B.: A shortest path-based approach to the multileaf collimator sequencing problem. Discrete Appl. Math. 160(1-2), 81–99 (2012)
Drucker, A.: New limits to classical and quantum instance compression. In: Proc. 53rd IEEE FOCS, pp. 609–618. IEEE Computer Society (2012)
Ehrgott, M., Güler, C., Hamacher, H., Shao, L.: Mathematical optimization in intensity modulated radiation therapy. Ann. Oper. Res. 175, 309–365 (2010)
Fellows, M.R., Koblitz, N.: Fixed-parameter complexity and cryptography. In: Moreno, O., Cohen, G., Mora, T. (eds.) AAECC 1993. LNCS, vol. 673, pp. 121–131. Springer, Heidelberg (1993)
Fellows, M.R., Gaspers, S., Rosamond, F.A.: Parameterizing by the number of numbers. Theory Comput. Syst. 50(4), 675–693 (2012)
Fellows, M.R., Jansen, B.M.P., Rosamond, F.A.: Towards fully multivariate algorithmics: Parameter ecology and the deconstruction of computational complexity. European J. Combin. 34(3), 541–566 (2013)
Frank, A., Tardos, É.: An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica 7(1), 49–65 (1987)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman (1979)
Guo, J., Hüffner, F., Niedermeier, R.: A structural view on parameterizing problems: Distance from triviality. In: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 162–173. Springer, Heidelberg (2004)
Kannan, R.: Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12, 415–440 (1987)
Karloff, H., Korn, F., Makarychev, K., Rabani, Y.: On parsimonious explanations for 2-d tree- and linearly-ordered data. In: Proc. 28th STACS. LIPIcs, vol. 9, pp. 332–343. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2011)
Luan, S., Saia, J., Young, M.: Approximation algorithms for minimizing segments in radiation therapy. Inf. Process. Lett. 101(6), 239–244 (2007)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press (2006)
Niedermeier, R.: Reflections on multivariate algorithmics and problem parameterization. In: Proc. 27th STACS. LIPIcs, vol. 5, pp. 17–32. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2010)
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Bredereck, R., Chen, J., Hartung, S., Komusiewicz, C., Niedermeier, R., Suchý, O. (2013). On Explaining Integer Vectors by Few Homogenous Segments. In: Dehne, F., Solis-Oba, R., Sack, JR. (eds) Algorithms and Data Structures. WADS 2013. Lecture Notes in Computer Science, vol 8037. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40104-6_18
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DOI: https://doi.org/10.1007/978-3-642-40104-6_18
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