The Complexity of Finding Tangles

We study the following combinatorial problem. Given a set of n y-monotone curves, which we call , a determines the order of the wires on a number of horizontal such that any two consecutive layers differ only in swaps of neighboring wires. Given a multiset L of (that is, unordered pairs of wires) and an initial order of the wires, a tangle  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.