Skip to main content
SpringerLink
Log in

Parameterizing by the Number of Numbers

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

The usefulness of parameterized algorithmics has often depended on what Niedermeier has called “the art of problem parameterization”. In this paper we introduce and explore a novel but general form of parameterization: the number of numbers. Several classic numerical problems, such as Subset Sum, Partition, 3-Partition, Numerical 3-Dimensional Matching, and Numerical Matching with Target Sums, have multisets of integers as input. We initiate the study of parameterizing these problems by the number of distinct integers in the input. We rely on an FPT result for Integer Linear Programming Feasibility to show that all the above-mentioned problems are fixed-parameter tractable when parameterized in this way. In various applied settings, problem inputs often consist in part of multisets of integers or multisets of weighted objects (such as edges in a graph, or jobs to be scheduled). Such number-of-numbers parameterized problems often reduce to subproblems about transition systems of various kinds, parameterized by the size of the system description. We consider several core problems of this kind relevant to number-of-numbers parameterization. Our main hardness result considers the problem: given a non-deterministic Mealy machine M (a finite state automaton outputting a letter on each transition), an input word x, and a census requirement c for the output word specifying how many times each letter of the output alphabet should be written, decide whether there exists a computation of M reading x that outputs a word y that meets the requirement c. We show that this problem is hard for W[1]. If the question is whether there exists an input word x such that a computation of M on x outputs a word that meets c, the problem becomes fixed-parameter tractable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Alon, N., Azar, Y., Woeginger, G.J., Yadid, T.: Approximation schemes for scheduling on parallel machines. J. Sched. 1, 55–66 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bard, G.V.: Algebraic Cryptanalysis. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  3. Bazgan, C.: Schémas d’approximation et complexité paramétrée. Master’s thesis, Université Paris Sud (1995)

  4. Betzler, N., Fellows, M.R., Guo, J., Niedermeier, R., Rosamond, F.A.: Fixed-parameter algorithms for Kemeny rankings. Theor. Comput. Sci. 410(45), 4554–4570 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chrobak, M., Dürr, C., Hurand, M., Robert, J.: Algorithms for temperature-aware task scheduling in microprocessor systems. In: Fleischer, R., Jinhui, X. (eds.) AAIM 2008. Lecture Notes in Computer Science, vol. 5034, pp. 120–130. Springer, Berlin (2008)

    Google Scholar 

  6. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)

    Book  Google Scholar 

  7. Fellows, M.R., Gaspers, S., Rosamond, F.A.: Parameterizing by the number of numbers. In: Raman, V., Saurabh, S. (eds.) IPEC 2010. Lecture Notes in Computer Science, vol. 6478, pp. 123–134. Springer, Berlin (2010)

    Google Scholar 

  8. Fellows, M.R., Hermelin, D., Rosamond, F., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci. 410(1), 53–61 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fellows, M.R., Koblitz, N.: Fixed-parameter complexity and cryptography. In: Cohen, G.D., Mora, T., Moreno, O. (eds.) AAECC 1993. Lecture Notes in Computer Science, vol. 673, pp. 121–131. Springer, Berlin (1993)

    Google Scholar 

  10. Fellows, M.R., Lokshtanov, D., Misra, N., Mnich, M., Rosamond, F.A., Saurabh, S.: The complexity ecology of parameters: An illustration using bounded max leaf number. Theory Comput. Syst. 45(4), 822–848 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fellows, M.R., Lokshtanov, D., Misra, N., Rosamond, E.A., Saurabh, S.: Graph layout problems parameterized by vertex cover. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. Lecture Notes in Computer Science, vol. 5369, pp. 294–305. Springer, Berlin (2008)

    Google Scholar 

  12. Fiala, J., Golovach, P.A., Kratochvíl, J.: Parameterized complexity of coloring problems: Treewidth versus vertex cover. Theor. Comput. Sci. 412(23), 2513–2523 (2011)

    Article  MATH  Google Scholar 

  13. Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series, vol. XIV. Springer, Berlin (2006)

    Google Scholar 

  14. Frank, A., Tardos, É.: An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica 7(1), 49–65 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gaspers, S., Liedloff, M., Stein, M.J., Suchan, K.: Complexity of splits reconstruction for low-degree trees. In: Kratochvíl, J. (ed.) WG 2011. Lecture Notes in Computer Science, vol. 6986 pp. 167–178. Springer, Berlin (2011)

    Google Scholar 

  16. Gramm, J., Niedermeier, R., Rossmanith, P.: Fixed-parameter algorithms for closest string and related problems. Algorithmica 37(1), 25–42 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kannan, R.: Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12(3), 415–440 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  18. Knuth, D.E.: The Art of Computer Programming, Volume III: Sorting and Searching. Addison-Wesley, Reading (1973)

    Google Scholar 

  19. Komusiewicz, C., Niedermeier, R., Uhlmann, J.: Deconstructing intractability—a multivariate complexity analysis of interval constrained coloring. J. Discrete Algorithms 9(1), 137–151 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lenstra, H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mealy, G.H.: A method for synthesizing sequential circuits. Bell Syst. Tech. J. 34(5), 1045–1079 (1955)

    MathSciNet  Google Scholar 

  22. Munro, I., Spira, P.M.: Sorting and searching in multisets. SIAM J. Comput. 5(1), 1–8 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  23. Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2006)

    Book  MATH  Google Scholar 

  24. Niedermeier, R.: Reflections on multivariate algorithmics and problem parameterization. In: Marion, J.-Y., Schwentick, T. (eds.) STACS 2010. LIPIcs, vol. 5, pp. 17–32. Schloss Dagstuhl—Leibniz-Zentrum fuer Informatik, Dagstuhl (2010)

    Google Scholar 

  25. Parikh, R.J.: On context-free languages. J. ACM 13(4), 570–581 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pietrzak, K.: On the parameterized complexity of the fixed alphabet shortest common supersequence and longest common subsequence problems. J. Comput. Syst. Sci. 67(4), 757–771 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Roche, E., Schabes, Y.: Finite-state language processing. The MIT Press, Cambridge (1997)

    Google Scholar 

  28. Savage, J.E.: Models of Computation—Exploring the Power of Computing. Addison-Wesley, Reading (1998)

    MATH  Google Scholar 

  29. Sen, S., Gupta, N.: Distribution-sensitive algorithms. Nord. J. Comput. 6, 194–211 (1999)

    MathSciNet  MATH  Google Scholar 

  30. Sontag, E.: Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edn. Springer, Berlin (1998)

    MATH  Google Scholar 

Download references

Acknowledgement

We thank Iyad Kanj for stimulating conversations about this work.

Author information

Authors and Affiliations

  1. School of Engineering and IT, Charles Darwin University, Darwin, NT, 0909, Australia

    Michael R. Fellows & Frances A. Rosamond

  2. Institute of Information Systems, Vienna University of Technology, Vienna, Austria

    Serge Gaspers

Authors
  1. Michael R. Fellows
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Serge Gaspers
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Frances A. Rosamond
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Serge Gaspers.

Additional information

A preliminary version of this paper appeared in the proceedings of IPEC 2010 [7].

M.R.F. and F.A.R. acknowledge support from the Australian Research Council. S.G. acknowledges partial support from the European Research Council (COMPLEX REASON, 239962), from Conicyt Chile (Basal-CMM), and from the Australian Research Council.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fellows, M.R., Gaspers, S. & Rosamond, F.A. Parameterizing by the Number of Numbers. Theory Comput Syst 50, 675–693 (2012). https://doi.org/10.1007/s00224-011-9367-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-011-9367-y

Keywords

Search

Navigation