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 even significantly improves the previously known upper bound for the unweigthed set splitting problem. We also show that our algorithm can be de-randomized, thus derive the first fixed parameter tractable algorithm for the weighted set splitting problem.
This work was supported in part by the National Science Foundation under the Grants CCR-0311590 and CCF-0430683.
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References
Alon, N., Babai, L., Itai, A.: A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms 7, 567–683 (1986)
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)
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer, Heidelberg (1999)
Chen, J., Kanj, I.: Improved Exact Algorithms for Max-Sat. Discrete Applied Mathematics 142, 17–27 (2004)
Chen, J., Lu, S., Sze, S., Zhang, F.: Improved algorithms for path, matching, and packing problems. In: SODA, pp. 298–307 (2007)
Dehne, F., Fellows, M., Rosamond, F.: An FPT Algorithm for Set Splitting. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 180–191. Springer, Heidelberg (2003)
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: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 127–137. Springer, Heidelberg (2004)
Downey, R., Fellows, M.: Parameterized Complexity. Springer, Heidelberg (1999)
Fredman, M., Komlos, J., Szemeredi, E.: Storing a sparse table with O(1) worst case access time. Journal of the ACM 31, 538–544 (1984)
Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)
Kann, V., Lagergren, J., Panconesi, A.: Approximability of maximum splitting of k-sets and some other apx-complete problems. Information Processing Letters 58(3), 105–110 (1996)
Kneis, J., Molle, D., Richter, S., Rossmanith, P.: Divide-and-color. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 58–67. Springer, Heidelberg (2006)
Liu, Y., Lu, S., Chen, J., Sze, S.: Greedy localization and color-coding: Improved Matching and Packing Algorithms. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 84–95. Springer, Heidelberg (2006)
Lokshtanov, D., Sloper, C.: Fixed parameter set splitting, linear kernel and improved running time. Algorithms and Complexity in Durham 2005, King’s College Press, Texts in Algorithmics 4, 105–113 (2005)
Naor, M., Schulman, L., Srinivasan, A.: Splitters and near-optimal derandomization. In: FOCS, pp. 182–190 (1995)
Zhang, H., Ling, C.: An improved learning algorithm for augmented naive bayes. In: Cheung, D., Williams, G.J., Li, Q. (eds.) PAKDD 2001. LNCS (LNAI), vol. 2035, pp. 581–586. Springer, Heidelberg (2001)
Zwick, U.: Approximation algorithms for constraint satisfaction problems involving at most three variables per constraint. In: SODA, pp. 201–220 (1998)
Zwick, U.: Outward rotations: A tool for rounding solutions of semidefinite programming relaxation, with applications to max cut and other problem. In: STOC, pp. 679–687 (1999)
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Chen, J., Lu, S. (2007). Improved Algorithms for Weighted and Unweighted Set Splitting Problems. In: Lin, G. (eds) Computing and Combinatorics. COCOON 2007. Lecture Notes in Computer Science, vol 4598. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73545-8_52
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DOI: https://doi.org/10.1007/978-3-540-73545-8_52
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