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.

翻译：NP难的多重击中问题要求找出一个最小基数的集合，该集合与给定输入集中的每个集合相交指定的次数。我们推广了Weihe提出的著名数据归约算法，给出了一个针对多重击中问题的核，其参数为Dilworth数——这是Foldes和Hammer于1978年引入的图参数，但似乎在参数化复杂性理论领域此前未被探索。通过运用矩阵乘法，我们将算法加速到二次序列时间和对数并行时间。我们通过实验评估了算法。通过在GPU上实现算法，我们证明了在SIMD（单指令多数据流）架构上实现核化算法的可行性。