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Improved Parameterized Set Splitting Algorithms: A Probabilistic Approach

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Abstract

In this paper, we study parameterized algorithms for the set splitting problem, for both weighted and unweighted versions. First, we develop a new and effective technique based on a probabilistic method that allows us to develop a simpler and more efficient deterministic kernelization algorithm for the unweighted set splitting problem. We then propose a randomized algorithm for the weighted set splitting problem that is based on a new subset partition technique and has its running time bounded by O *(2k), which is significantly better than that of the previous best deterministic algorithm (which only works for the simpler unweighted set splitting problem) of running time O *(2.65k). We also show that our algorithm can be de-randomized, which leads to a deterministic parameterized algorithm of running time O *(4k) for the weighted set splitting problem and gives the first proof that the problem is fixed-parameter tractable.

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References

  1. Alon, N., Babai, L., Itai, A.: A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithms 7, 567–683 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  2. Andersson, G., Engebretsen, L.: Better approximation algorithms and tighter analysis for set splitting and not-all-equal Sat. In: ECCCTR: Electronic Colloquium on Computational Complexity (1997)

  3. Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer, Berlin (1999)

    MATH  Google Scholar 

  4. Chen, J., Kanj, I.: Improved exact algorithms for Max-Sat. Discrete Appl. Math. 142, 17–27 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen, J., Lu, S.: Improved algorithm for weighted and unweighted set splitting problems. In: COCOON 2007. Lecture Notes in Computer Science, vol. 4598, pp. 573–547 (2007)

  6. Chen, J., Lu, S., Sze, S., Zhang, F.: Improved algorithms for path, matching, and packing problems. In: Proc. of the Eighteen Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), pp. 298–307 (2007)

  7. Dehne, F., Fellows, M., Rosamond, F.: An FPT algorithm for set splitting. In: WG 2003. Lecture Notes in Computer Science, vol. 2880, pp. 180–191 (2003)

  8. Dehne, F., Fellows, M., Rosamond, F., Shaw, P.: Greedy localization, iterative compression, modeled crown reductions: New FPT techniques, and improved algorithm for set splitting, and a novel 2k kernelization of vertex cover. In: IWPEC 2004. Lecture Notes in Computer Science, vol. 3162, pp. 127–137 (2004)

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

    Google Scholar 

  10. Fredman, M., Komlos, J., Szemeredi, E.: Storing a sparse table with O(1) worst case access time. J. ACM 31, 538–544 (1984)

    MATH  MathSciNet  Google Scholar 

  11. Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)

    MATH  Google Scholar 

  12. Hardy, G., Wright, E.: An Introduction to the Theory of Numbers, 5th ed. Oxford University Press, London (1978)

    Google Scholar 

  13. Kann, V., Lagergren, J., Panconesi, A.: Approximability of maximum splitting of k-sets and some other APX-complete problems. Inf. Process. Lett. 58, 105–110 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kneis, J., Mölle, D., Richter, S., Rossmanith, P.: Divide-and-color. In: WG 2006. Lecture Notes in Computer Science, vol. 4271, pp. 58–67 (2006)

  15. Lokshtanov, D., Sloper, C.: Fixed parameter set splitting, linear kernel and improved running time. In: Algorithms and Complexity in Durham 2005. Texts in Algorithmics, vol. 4, pp. 105–113. King’s College Press, London (2005)

    Google Scholar 

  16. Naor, M., Schulman, L., Srinivasan, A.: Splitters and near-optimal derandomization. In: Proc. 36th IEEE Symp. on Foundations of Computer Science (FOCS 1995), pp. 182–190 (1995)

  17. Zhang, H., Ling, C.: An improved learning algorithm for augmented naive Bayes. In: PAKDD 2001. Lecture Notes in Computer Science, vol. 2035, pp. 581–586 (2001)

  18. Zwick, U.: Approximation algorithms for constraint satisfaction problems involving at most three variables per constraint. In: Proc. of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1998), pp. 201–220 (1998)

  19. Zwick, U.: Outward rotations: A tool for rounding solutions of semidefinite programming relaxation, with applications to max cut and other problem. In: Proc. of the Thirty-First Annual ACM Symposium on Theory of Computing (STOC 1999), pp. 679–687 (1999)

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Authors and Affiliations

  1. Department of Computer Science, Texas A&M University, College Station, TX, 77843, USA

    Jianer Chen & Songjian Lu

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  1. Jianer Chen
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  2. Songjian Lu
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Correspondence to Songjian Lu.

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A preliminary version of this paper was presented at The 13th Annual International Computing and Combinatorics Conference (COCOON 2007), Banff, Canada, July 2007, LNCS vol. 4598, pp. 537–547.

This work was supported in part by the National Science Foundation under the Grant CCF-0430683.

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Chen, J., Lu, S. Improved Parameterized Set Splitting Algorithms: A Probabilistic Approach. Algorithmica 54, 472–489 (2009). https://doi.org/10.1007/s00453-008-9206-y

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