并行计算内核：下限和上限
Computing Kernels in Parallel: Lower and Upper Bounds
摘要：
并行固定参数易处理性研究如何并行解决参数化问题。令人惊讶的大量参数化问题允许高水平的并行化，但这并不意味着我们也可以有效地并行计算小问题核：已知的核化算法通常是高度顺序的。在本文中，我们建立了一些上限和下限的大小的内核，可以并行计算。一个有趣的发现是，有复杂的权衡核的大小和深度的电路需要计算它们：对于顶点覆盖问题，指数核可以计算AC $^0$ -电路，二次核TC $^0$ -电路，和线性核随机NC-电路与去随机化是可能的，只有当它也是可能的匹配问题。其他自然的问题，类似的（但数量上不同）的影响，可以观察到包括树分解问题参数化的顶点覆盖数，无向反馈顶点集问题，匹配问题，或点线覆盖问题。我们还提出了自然的问题，计算内核是固有的顺序。
英文摘要：
Parallel fixed-parameter tractability studies how parameterized problems can be solved in parallel. A surprisingly large number of parameterized problems admit a high level of parallelization, but this does not mean that we can also efficiently compute small problem kernels in parallel: known kernelization algorithms are typically highly sequential. In the present paper, we establish a number of upper and lower bounds concerning the sizes of kernels that can be computed in parallel. An intriguing finding is that there are complex trade-offs between kernel size and the depth of the circuits needed to compute them: For the vertex cover problem, an exponential kernel can be computed by AC$^0$-circuits, a quadratic kernel by TC$^0$-circuits, and a linear kernel by randomized NC-circuits with derandomization being possible only if it is also possible for the matching problem. Other natural problems for which similar (but quantitatively different) effects can be observed include tree decomposition problems parameterized by the vertex cover number, the undirected feedback vertex set problem, the matching problem, or the point line cover problem. We also present natural problems for which computing kernels is inherently sequential.
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