Fixed-parameter algorithms and kernelization are two powerful methods to solve NP-hard problems. Yet so far those algorithms have been largely restricted to static inputs. In this article, we provide fixed-parameter algorithms and kernelizations for fundamental NP-hard problems with dynamic inputs. We consider a variety of parameterized graph and hitting set problems that are known to have f ( k ) n 1+o(1) time algorithms on inputs of size n , and we consider the question of whether there is a data structure that supports small updates (such as edge/vertex/set/element insertions and deletions) with an update time of g ( k ) n o(1) ; such an update time would be essentially optimal. Update and query times independent of n are particularly desirable. Among many other results, we show that F EEDBACK V ERTEX S ET and k -P ATH admit dynamic algorithms with f ( k )log O(1) update and query times for some function f depending on the solution size k only. We complement our positive results by several conditional and unconditional lower bounds. For example, we show that unlike their undirected counterparts, D IRECTED F EEDBACK V ERTEX S ET and D IRECTED k -P ATH do not admit dynamic algorithms with n o(1) update and query times even for constant solution sizes k ≤ 3 , assuming popular hardness hypotheses. We also show that unconditionally, in the cell probe model, D IRECTED F EEDBACK V ERTEX S ET cannot be solved with update time that is purely a function of k .

中文翻译：

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动态参数化问题和算法

固定参数算法和核化是解决 NP-hard 问题的两种强有力的方法。然而到目前为止，这些算法在很大程度上仅限于静止的输入。在本文中，我们为基本的 NP-hard 问题提供了固定参数算法和核化动态的输入。我们考虑了各种已知的参数化图和命中集问题F(ķ)n 1+o(1)大小输入的时间算法n，并且我们考虑是否存在支持更新时间为G(ķ)n o(1); 这样的更新时间基本上是最佳的。更新和查询时间独立于n是特别可取的。在许多其他结果中，我们表明 F回馈五ERTEX小号外星人和ķ-PATH承认动态算法F(ķ）日志O(1)某些功能的更新和查询时间F取决于解决方案的大小ķ只要。我们通过几个有条件和无条件的下限来补充我们的积极结果。例如，我们表明，与它们的无向对应物不同，D已收到F回馈五ERTEX小号外星人和 D已收到 ķ-PATH不接受具有 n 的动态算法o(1)即使对于恒定的解决方案大小，也可以更新和查询时间ķ≤3，假设流行的硬度假设。我们还无条件地表明，在细胞探针模型中，D已收到F回馈五ERTEX小号外星人不能用纯粹是函数的更新时间来解决ķ.