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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- 1.
This is usually called a (disjunctive) Turing kernelization.
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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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