Algorithms and Complexity for Turaev-Viro Invariants

Abstract

The Turaev-Viro invariants are a powerful family of topological invariants for distinguishing between different 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them require exponential time.The invariants are parameterised by an integer r ≥ 3. We resolve the question of complexity for r = 3 and r = 4, giving simple proofs that computing Turaev-Viro invariants for r = 3 is polynomial time, but for r = 4 is #P-hard. Moreover, we give an explicit fixed-parameter tractable algorithm for arbitrary r, and show through concrete implementation and experimentation that this algorithm is practical-and indeed preferableto the prior state of the art for real computation.

On the treewidth of triangulated 3-manifolds

“…Let T {1,3} be the set of unrooted binary trees. 17 The congestion cng(G) of a graph G is defined as…”

“…In recent years, there have been several attempts to explain this gap using the concepts of parameterized complexity and algorithms for fixed parameter tractable (FPT) problems [23,24]. This effort has proven to be highly effective and, today, there exist numerous FPT algorithms in the field [15,16,17,18,44]. More specifically, given a triangulation T of a 3-manifold M with n tetrahedra whose dual graph Γ(T ) has treewidth 2 at most k, there exist algorithms to compute • taut angle structures 3 of what is called ideal triangulations with torus boundary components in running time O(7 k ⋅ n) [18];…”

“…As a result, they have a significant practical impact. This is in particular the case for the algorithm to compute Turaev-Viro invariants [17,44].…”

“…By a theorem due to Kirby and Melvin [37], the Turaev-Viro invariant for parameter r = 4 is #P-hard to compute. Thus, if there were a polynomial time procedure to turn an n-tetrahedron triangulation T into a poly(n)-tetrahedron triangulation T ′ with dual graph of treewidth at most k, for some universal constant k, then this procedure, combined with the algorithm from [17], would constitute a polynomial time solution for a #P-hard problem. Furthermore, known facts imply that most triangulations of most 3-manifolds must have large treewidth 7 (see Propositions 22 and 23 in Appendix A).…”

The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex

“…Our work is one of the few ones combining topology and fixed parameter tractability. In this direction there have been recent results focused on algorithms in 3-manifold topology [2,5,6,7,16]. The problem of finding a shortest 1-dimensional cycle Z 2homologous to a given cycle in a 2-dimensional cycle was shown to be NP-hard by Chao and Freedman [9].…”