Parameterized complexity of untangling knots
Computational Complexity
2021-11-10 v1 Computational Geometry
Geometric Topology
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 the II- moves in a shortest untangling sequence 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.
Cite
@article{arxiv.2111.05001,
title = {Parameterized complexity of untangling knots},
author = {Clément Legrand-Duchesne and Ashutosh Rai and Martin Tancer},
journal= {arXiv preprint arXiv:2111.05001},
year = {2021}
}