Deciding the Feasibility and Minimizing the Height of Tangles

We study the following combinatorial problem. Given a set of n y-monotone wires, a tangle determines the order of the wires on a number of horizontal layers such that the orders of the wires on 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. List-Feasibility is the problem of finding a tangle that realizes a given list L if such a tangle exists. Tangle-Height Minimization is the problem of finding a tangle that realizes a given list and additionally uses the minimum number of layers. List-Feasibility (and therefore Tangle-Height Minimization) is NP-hard [Yamanaka, Horiyama, Uno, Wasa; CCCG 2018]. We prove that List-Feasibility remains NP-hard if every pair of wires swaps only a constant number of times. On the positive side, we present an algorithm for Tangle-Height Minimization that computes an optimal tangle for n wires and a given list L of swaps in O((2|L|/n^2+1)^n^2/2·φ^n · n) time, where φ≈ 1.618 is the golden ratio and |L| is the total number of swaps in L. From this algorithm, we derive a simpler and faster version to solve List-Feasibility. We also use the algorithm to show that List-Feasibility is in NP and fixed-parameter tractable with respect to the number of wires. For simple lists, where every swap occurs at most once, we show how to solve Tangle-Height Minimization in O(n!φ^n) time.