Parameterized complexity of untangling knots

Clément Legrand-Duchesne; Ashutosh Rai; Martin Tancer

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arXiv · 2111.05001

Parameterized complexity of untangling knots

Clément Legrand-Duchesne·Ashutosh Rai·Martin Tancer

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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.

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BibTeX RIS

Clément Legrand-Duchesne, Ashutosh Rai, Martin Tancer. 2021-11-09. Parameterized complexity of untangling knots. https://arxiv.org/abs/2111.05001

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