We study the following combinatorial problem. Given a set of $n$ y-monotone curves, which we call wires, a tangle determines the order of the wires on a number of horizontal layers such that any two consecutive layers differ only in swaps of neighboring wires. Given a multiset $L$ of swaps (that is, unordered pairs of wires) and an initial order of the wires, a tangle realizes $L$ if each pair of wires changes its order exactly as many times as specified by $L$. Deciding whether a given multiset of swaps admits a realizing tangle is known to be NP-hard [Yamanaka et al., CCCG 2018]. We prove that this problem remains NP-hard if every pair of wires swaps only a constant number of times. On the positive side, we improve the runtime of a previous exponential-time algorithm. We also show that the problem is in NP and fixed-parameter tractable with respect to the number of wires.

翻译：我们研究以下组合问题：给定一组$n$条$y$单调曲线（称为导线），一个缠结决定了这些导线在若干水平层上的顺序，使得任意相邻两层之间仅通过交换相邻导线来改变顺序。给定一个交换对（即导线的无序对）的多重集$L$以及导线的初始顺序，若缠结实现了$L$，则每对导线交换顺序的次数恰好等于$L$中指定的次数。判断给定的交换多重集是否存在可实现缠结的问题已知为NP难问题[Yamanaka等，CCCG 2018]。我们证明，即使每对导线仅交换常数次，该问题仍为NP难问题。在正面结果方面，我们改进了先前指数时间算法的运行时间。我们还证明该问题属于NP类，并且关于导线数量是固定参数可解的。