Parameterized Complexity of Untangling Knots

IF 1.2 3区计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS

SIAM Journal on Computing Pub Date : 2024-03-21 DOI:10.1137/22m1501969

Clément Legrand-Duchesne, Ashutosh Rai, Martin Tancer

引用次数: 0

Abstract

SIAM Journal on Computing, Volume 53, Issue 2, Page 431-479, April 2024.
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 in a shortest untangling sequence the [math] moves, that is, the moves removing two adjacent crossings, 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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解结的参数化复杂性

SIAM 计算期刊》，第 53 卷第 2 期，第 431-479 页，2024 年 4 月。摘要。已知用给定的移动次数（作为输入的一部分）决定一个结的图是否能解开是一个 NP-完全问题。在本文中，我们确定了这个问题的参数化复杂度，它与一个称为缺陷的自然参数有关。粗略地说，它衡量的是莱德米斯特移动最短解缠序列中所用移动的效率。我们证明，在最短的解缠序列中，[math] 移动，即去除两个相邻交叉点的移动，基本上可以贪婪地进行。据此，我们证明当以缺陷为参数时，这个问题属于 W[P]。通过最小公理集的还原，我们还证明这个问题是 W[P] 难问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Computing 工程技术-计算机：理论方法

CiteScore

4.60

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Computing aims to provide coverage of the most significant work going on in the mathematical and formal aspects of computer science and nonnumerical computing. Submissions must be clearly written and make a significant technical contribution. Topics include but are not limited to analysis and design of algorithms, algorithmic game theory, data structures, computational complexity, computational algebra, computational aspects of combinatorics and graph theory, computational biology, computational geometry, computational robotics, the mathematical aspects of programming languages, artificial intelligence, computational learning, databases, information retrieval, cryptography, networks, distributed computing, parallel algorithms, and computer architecture.