Computing higher homotopy groups is W[1]-hard
Computational Complexity
2013-04-30 v1 Computational Geometry
Abstract
Recently it was shown that, for every fixed k>1, given a finite simply connected simplicial complex X, the kth homotopy group \pi_k(X) can be computed in time polynomial in the number n of simplices of X. We prove that this problem is W[1]-hard w.r.t. the parameter k even for X of dimension 4, and thus very unlikely to admit an algorithm with running time bound f(k)n^C for an absolute constant C. We also simplify, by about 20 pages, a 1989 proof by Anick that, with k part of input, the computation of the rank of \pi_k(X) is #P-hard.
Keywords
Cite
@article{arxiv.1304.7705,
title = {Computing higher homotopy groups is W[1]-hard},
author = {Jiri Matousek},
journal= {arXiv preprint arXiv:1304.7705},
year = {2013}
}