Abstract
Kernelization is a formalization of efficient preprocessing for \(\mathsf {NP}\)-hard problems using the framework of parameterized complexity. Among open problems in kernelization it has been asked many times whether there are deterministic polynomial kernelizations for Subset Sum and Knapsack when parameterized by the number n of items.
We answer both questions affirmatively by using an algorithm for compressing numbers due to Frank and Tardos (Combinatorica 1987). This result had been first used by Marx and Végh (ICALP 2013) in the context of kernelization. We further illustrate its applicability by giving polynomial kernels also for weighted versions of several well-studied parameterized problems. Furthermore, when parameterized by the different item sizes we obtain a polynomial kernelization for Subset Sum and an exponential kernelization for Knapsack. Finally, we also obtain kernelization results for polynomial integer programs.
Supported by the Emmy Noether-program of the German Research Foundation (DFG), KR 4286/1, and ERC Starting Grant 306465 (BeyondWorstCase).
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Notes
- 1.
This is usually called a (disjunctive) Turing kernelization.
References
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)
Bodlaender, H.L., Fomin, F.V., Saurabh, S.: Open problems posed at WORKER 2010 (2010). http://fpt.wdfiles.com/local-files/open-problems/open-problems.pdf
Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Kernelization lower bounds by cross-composition. SIAM J. Discrete Math. 28(1), 277–305 (2014)
Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. Theoret. Comput. Sci. 412(35), 4570–4578 (2011)
Chlebík, M., Chlebíková, J.: Crown reductions for the minimum weighted vertex cover problem. Discrete Appl. Math. 156(3), 292–312 (2008)
Cygan, M., Fomin, F.V., Jansen, B.M.P., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Open problems for FPT school 2014 (2014). http://fptschool.mimuw.edu.pl/opl.pdf
Cygan, M., Kowalik, L., Pilipczuk, M.: Open problems from workshop on kernels (2013). http://worker2013.mimuw.edu.pl/slides/worker-opl.pdf
Drucker, A.: New limits to classical and quantum instance compression. In: Proceedings of the FOCS 2012, pp. 609–618 (2012)
Erdős, P., Rado, R.: Intersection theorems for systems of sets. J. London Math. Soc. 35, 85–90 (1960)
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., Guo, J., Marx, D., Saurabh, S.: Data reduction and problem kernels (dagstuhl seminar 12241). Dagstuhl Rep. 2(6), 26–50 (2012)
Frank, A., Tardos, É.: An application of simultaneous Diophantine approximation in combinatorial optimization. Combinatorica 7(1), 49–65 (1987)
Granot, F., Skorin-Kapov, J.: On simultaneous approximation in quadratic integer programming. Oper. Res. Lett. 8(5), 251–255 (1989)
Harnik, D., Naor, M.: On the compressibility of \({\sf NP}\) instances and cryptographic applications. SIAM J. Comput. 39(5), 1667–1713 (2010)
Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity. J. Comput. Syst. Sci. 63(4), 512–530 (2001)
Jansen, K., Kratsch, S., Marx, D., Schlotter, I.: Bin packing with fixed number of bins revisited. J. Comput. Syst. Sci. 79(1), 39–49 (2013)
Kannan, R.: Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12(3), 415–440 (1987)
Kratsch, S.: On polynomial kernels for integer linear programs: covering, packing and feasibility. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 647–658. Springer, Heidelberg (2013)
Lenstra Jr, H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)
Marx, D., Végh, L.A.: Fixed-parameter algorithms for minimum cost edge-connectivity augmentation. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 721–732. Springer, Heidelberg (2013)
Nederlof, J., van Leeuwen, E.J., van der Zwaan, R.: Reducing a target interval to a few exact queries. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 718–727. Springer, Heidelberg (2012)
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Etscheid, M., Kratsch, S., Mnich, M., Röglin, H. (2015). Polynomial Kernels for Weighted Problems. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_24
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