Given a hypergraph H = (V,E), what is the smallest subset \(X \subseteq V\) such that e ∩ X≠∅ holds for all e ∈ E? This problem, known as the hitting set problem, is a basic problem in combinatorial optimization and has been studied extensively in both classical and parameterized complexity theory. There are well-known kernelization algorithms for it, which get a hypergraph H and a number k as input and output a hypergraph H^′ such that (1) H has a hitting set of size k if and only if \(H^{\prime }\) has such a hitting set and (2) the size of \(H^{\prime }\) depends only on k and on the maximum cardinality d of hyperedges in H. The algorithms run in polynomial time and can be parallelized to a certain degree: one can easily compute hitting set kernels in parallel time O(k) and not-so-easily in time O(d) – but it was conjectured that these are the best parallel algorithms possible. We refute this conjecture and show how hitting set kernels can be computed in constant parallel time. For our proof, we introduce a new, generalized notion of hypergraph sunflowers and show how iterated applications of the color coding technique can sometimes be collapsed into a single application.

中文翻译：

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通过AC 0-电路计算击中集合内核

鉴于超图^ h =（V，ē），什么是最小的子集\（X \ subseteq V \） ，使得Ë ∩ X ≠ ∅适用于所有Ë ∈ é？这个问题称为命中集问题，是组合优化中的一个基本问题，并且在经典和参数化复杂性理论中都进行了广泛研究。有众所周知的内核化算法，它获得一个超图H和一个数k作为输入并输出一个超图H ^′，使得（1）H具有击中集大小的ķ当且仅当\（H ^ {\素} \）具有这样的击中组和（2）的大小\（H ^ {\素} \）仅取决于ķ和上H中超边的最大基数d。该算法以多项式时间运行，并且可以在一定程度上并行化：人们可以轻松地在并行时间O（k）而不是在时间O（d）中轻易地计算命中集合内核–但据推测，这些是最好的并行算法。我们驳斥了这个猜想，并展示了如何以常数计算命中集内核平行时间。为了证明这一点，我们引入了一种新的广义的超图向日葵概念，并说明了有时如何将颜色编码技术的迭代应用折叠为单个应用。