SIAM Journal on Computing, Volume 53, Issue 2, Page 431-479, April 2024.
Abstract. Deciding whether a diagram of a knot can be untangled with a given number of moves (as a part of the input) is known to be NP-complete. In this paper we determine the parameterized complexity of this problem with respect to a natural parameter called defect. Roughly speaking, it measures the efficiency of the moves used in the shortest untangling sequence of Reidemeister moves. We show that in a shortest untangling sequence the [math] moves, that is, the moves removing two adjacent crossings, can be essentially performed greedily. Using that, we show that this problem belongs to W[P] when parameterized by the defect. We also show that this problem is W[P]-hard by a reduction from Minimum axiom set.

中文翻译：

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解开结的参数化复杂性

SIAM 计算杂志，第 53 卷，第 2 期，第 431-479 页，2024 年 4 月。
摘要。确定结的图是否可以通过给定数量的移动（作为输入的一部分）来解开已知是 NP 完全的。在本文中，我们根据称为缺陷的自然参数确定该问题的参数化复杂性。粗略地说，它衡量了雷德迈斯特动作的最短解开序列中使用的动作的效率。我们证明，在最短的理清序列中，[数学]移动，即删除两个相邻交叉点的移动，本质上可以贪婪地执行。使用它，我们表明当由缺陷参数化时，这个问题属于 W[P]。我们还通过从最小公理集减少来证明这个问题是 W[P]-困难的。