For many fundamental problems in computational topology, such as unknot recognition and $3$-sphere recognition, the existence of a polynomial-time solution remains unknown. A major algorithmic tool behind some of the best known algorithms for these problems is normal surface theory. However, we currently have a poor understanding of the computational complexity of problems in normal surface theory: many such problems are still not known to have polynomial-time algorithms, yet proofs of $\mathrm{NP}$-hardness also remain scarce. We give three results that provide some insight on this front. A number of modern normal surface theoretic algorithms depend critically on the operation of finding a non-trivial normal sphere or disc in a $3$-dimensional triangulation. We formulate an abstract problem that captures the algebraic and combinatorial aspects of this operation, and show that this abstract problem is $\mathrm{NP}$-complete. Assuming $\mathrm{P}\neq\mathrm{NP}$, this result suggests that any polynomial-time procedure for finding a non-trivial normal sphere or disc will need to exploit some geometric or topological intuition. Another key operation, which applies to a much wider range of topological problems, involves finding a vertex normal surface of a certain type. We study two closely-related problems that can be solved using this operation. For one of these problems, we give a simple alternative solution that runs in polynomial time; for the other, we prove $\mathrm{NP}$-completeness.
翻译:对于计算表层的许多根本问题,比如 unknot 识别和 $$$- great- hardity 的证据也仍然很少。 我们给出了三种能提供这方面一些洞察力的结果。 一些现代普通表面理论算法的运行, 关键取决于如何在300美元维度的三角中找到一个非三维的正常球体或盘体。 然而, 我们目前对正常表面理论中问题的计算复杂性理解不甚清楚: 许多这样的问题仍然不为人知, 仍然无法知道有多元时算法, 但是对于 $\ mathrm{ NAP}$- hardity 的证据也仍然很少。 我们给出了三种结果, 能够提供这方面的一些洞察力。 一些现代正常表面理论算法的算法性算法, 一个简单的数学算法, 一个在最上一级或最上一级 的地平面的解算法, 需要一种最上面的地平面的解算法 。