On the hardness of finding normal surfaces
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2019
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914705782603776 |
|---|---|
| author | Burton, Benjamin A. He, Alexander |
| author_facet | Burton, Benjamin A. He, Alexander |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_09051 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On the hardness of finding normal surfaces Burton, Benjamin A. He, Alexander Computational Geometry Computational Complexity 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. |
| title | On the hardness of finding normal surfaces |
| topic | Computational Geometry Computational Complexity |
| url | https://arxiv.org/abs/1912.09051 |