The NP-hard Multiple Hitting Set problem is finding a minimum-cardinality set intersecting each of the sets in a given input collection a given number of times. Generalizing a well-known data reduction algorithm due to Weihe, we show a problem kernel for Multiple Hitting Set parameterized by the Dilworth number, a graph parameter introduced by Foldes and Hammer in 1978 yet seemingly so far unexplored in the context of parameterized complexity theory. Using matrix multiplication, we speed up the algorithm to quadratic sequential time and logarithmic parallel time. We experimentally evaluate our algorithms. By implementing our algorithm on GPUs, we show the feasability of realizing kernelization algorithms on SIMD (Single Instruction, Multiple Date) architectures.

中文翻译：

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由 Dilworth 数参数化的多重命中集的串行和并行内核化，在 GPU 上实现

NP-hard Multiple Hitting Set 问题是找到一个最小基数集，它与给定输入集合中的每个集相交给定次数。概括了由于 Weihe 著名的数据缩减算法，我们展示了由 Dilworth 数参数化的多重命中集的问题内核，Dilworth 数是一个由 Foldes 和 Hammer 在 1978 年引入的图参数，但目前似乎尚未在参数化复杂性理论的背景下进行探索。使用矩阵乘法，我们将算法加速到二次顺序时间和对数并行时间。我们通过实验评估我们的算法。通过在 GPU 上实现我们的算法，我们展示了在 SIMD（单指令、多数据）架构上实现内核化算法的可行性。