Reducing a Target Interval to a Few Exact Queries

  • Jesper Nederlof
  • Erik Jan van Leeuwen
  • Ruben van der Zwaan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)

Abstract

Many combinatorial problems involving weights can be formulated as a so-called ranged problem. That is, their input consists of a universe U, a (succinctly-represented) set family \(\mathcal{F} \subseteq 2^{U}\), a weight function ω:U → {1,…,N}, and integers 0 ≤ l ≤ u ≤ ∞. Then the problem is to decide whether there is an \(X \in \mathcal{F}\) such that l ≤ ∑ e ∈ Xω(e) ≤ u. Well-known examples of such problems include Knapsack, Subset Sum, Maximum Matching, and Traveling Salesman. In this paper, we develop a generic method to transform a ranged problem into an exact problem (i.e. a ranged problem for which l = u). We show that our method has several intriguing applications in exact exponential algorithms and parameterized complexity, namely:
  • In exact exponential algorithms, we present new insight into whether Subset Sum and Knapsack have efficient algorithms in both time and space. In particular, we show that the time and space complexity of Subset Sum and Knapsack are equivalent up to a small polynomial factor in the input size. We also give an algorithm that solves sparse instances of Knapsack efficiently in terms of space and time.

  • In parameterized complexity, we present the first kernelization results on weighted variants of several well-known problems. In particular, we show that weighted variants of Vertex Cover and Dominating Set, Traveling Salesman, and Knapsack all admit polynomial randomized Turing kernels when parameterized by |U|.

Curiously, our method relies on a technique more commonly found in approximation algorithms.

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References

  1. 1.
    Beier, R., Vöcking, B.: Random knapsack in expected polynomial time. J. Comput. Syst. Sci. 69(3), 306–329 (2004)MATHCrossRefGoogle Scholar
  2. 2.
    Bellman, R.E.: Dynamic Programming (reprint 2003). Dover Publications, Incorporated (1954)Google Scholar
  3. 3.
    Björklund, A.: Determinant sums for undirected Hamiltonicity. In: FOCS, pp. 173–182. IEEE Computer Society (2010)Google Scholar
  4. 4.
    Bodlaender, H.L.: Kernelization: New Upper and Lower Bound Techniques. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 17–37. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Fernau, H., Fomin, F.V., Lokshtanov, D., Raible, D., Saurabh, S., Villanger, Y.: Kernel(s) for problems with no kernel: On out-trees with many leaves. In: Albers, S., Marion, J.-Y. (eds.) STACS. LIPIcs, vol. 3, pp. 421–432. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany (2009)Google Scholar
  7. 7.
    Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms, 1st edn. Springer, New York (2010)MATHCrossRefGoogle Scholar
  8. 8.
    Harnik, D., Naor, M.: On the compressibility of np instances and cryptographic applications. SIAM J. Comput. 39(5), 1667–1713 (2010)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Ibarra, O., Kim, C.: Fast approximation algorithms for the knapsack and sum of subset problems. Journal of the ACM 22, 463–468 (1975)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Kaski, P., Koivisto, M., Nederlof, J.: Homomorphic hashing for sparse coefficient extraction. Manuscript (2012)Google Scholar
  11. 11.
    Lokshtanov, D., Nederlof, J.: Saving space by algebraization. In: Schulman, L.J. (ed.) STOC, pp. 321–330. ACM (2010)Google Scholar
  12. 12.
    Mansour, Y.: Randomized interpolation and approximation of sparse polynomials. SIAM J. Comput. 24(2), 357–368 (1995)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Nederlof, J.: Space and Time Efficient Structural Improvements of Dynamic Programming Algorithms. PhD thesis, University of Bergen (2011)Google Scholar
  14. 14.
    Nemhauser, G.L., Ullmann, Z.: Discrete dynamic programming and capital allocation. Management Science 15(9), 494–505 (1969)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Schroeppel, R., Shamir, A.: A T = O(2n/2), S = O(2n/4) algorithm for certain NP-complete problems. SIAM J. Comput. 10(3), 456–464 (1981)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Spielman, D.A., Teng, S.-H.: Smoothed analysis: an attempt to explain the behavior of algorithms in practice. Commun. ACM 52(10), 76–84 (2009)CrossRefGoogle Scholar
  17. 17.
    Woeginger, G.J.: Open problems around exact algorithms. Discrete Applied Mathematics 156(3), 397–405 (2008)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jesper Nederlof
    • 1
  • Erik Jan van Leeuwen
    • 2
  • Ruben van der Zwaan
    • 3
  1. 1.Utrecht UniversityThe Netherlands
  2. 2.Sapienza University of RomeItaly
  3. 3.Maastricht UniversityThe Netherlands

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