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On the hardness of finding normal surfaces

Benjamin A. Burton, Alexander He

TL;DR

Any polynomial-time procedure for finding a non-trivial normal sphere or disc in a 3-dimensional triangulation will need to exploit some geometric or topological intuition.

Abstract

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.

On the hardness of finding normal surfaces

TL;DR

Any polynomial-time procedure for finding a non-trivial normal sphere or disc in a 3-dimensional triangulation will need to exploit some geometric or topological intuition.

Abstract

For many fundamental problems in computational topology, such as unknot recognition and -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 -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 -dimensional triangulation. We formulate an abstract problem that captures the algebraic and combinatorial aspects of this operation, and show that this abstract problem is -complete. Assuming , 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 -completeness.

Paper Structure

This paper contains 6 sections, 8 theorems, 6 equations, 12 figures.

Table of Contents

  1. Keywords
  2. Introduction
  3. Preliminaries
  4. Abstract normal constraint optimisation
  5. Detecting splitting surfaces and connected spanning central surfaces
  6. Discussion

Key Result

Theorem 1

Let $\mathcal{T}$ be an $n$-tetrahedron triangulation. A vector $\mathbf{x}\in\mathbb{Z}^{7n}$ represents a normal surface in $\mathcal{T}$ (uniquely up to normal isotopy) if and only if $\mathbf{x}$ is admissible Haken61HassLagariasPippenger99.

Figures (12)

  • Figure 1: The seven elementary disc types.
  • Figure 2: The matching equations assert that the number of red arcs should equal the number of blue arcs.
  • Figure 3: Building a vertex-linking normal surface $S$ from triangles.
  • Figure 4: An elementary disc cuts a tetrahedron into two regions.
  • Figure 5: Construction of the triangular solid torus.
  • ...and 7 more figures

Theorems & Definitions (8)

  • Theorem 1: Haken
  • Lemma 3
  • Lemma 6
  • Lemma 7
  • Theorem 8
  • Proposition 11
  • Theorem 12
  • Theorem 18