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A kernelization algorithm for $$d$$-hitting set. (English) Zbl 1197.68083
Summary: For a given parameterized problem, $$\pi$$, a kernelization algorithm is a polynomial-time pre-processing procedure that transforms an arbitrary instance of $$\pi$$ into an equivalent one whose size depends only on the input parameter(s). The resulting instance is called a problem kernel. In this paper, a kernelization algorithm for the 3-Hitting Set problem is presented along with a general kernelization for $$d$$-Hitting Set. For 3-Hitting Set, an arbitrary instance is reduced into an equivalent one that contains at most $$5k^{2}+k$$ elements. This kernelization is an improvement over previously known methods that guarantee cubic-order kernels. Our method is used also to obtain quadratic kernels for several other problems. For a constant $$d \geqslant 3$$, a kernelization of $$d$$-Hitting Set is achieved by a non-trivial generalization of the 3-Hitting Set method, and guarantees a kernel whose order does not exceed $$(2d - 1)k^{d - 1}+k$$.

##### MSC:
 68W05 Nonnumerical algorithms 68Q25 Analysis of algorithms and problem complexity 68R10 Graph theory (including graph drawing) in computer science
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##### References:
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