VEST is W[2]-hard

TL;DR

This paper proves that the VEST problem is W[2]-hard for parameter k, indicating increased computational complexity and extending previous results that showed W[1]-hardness, with implications for computing homotopy groups.

Contribution

It establishes the W[2]-hardness of VEST, strengthening prior W[1]-hardness results and impacting the complexity understanding of homotopy group computations.

Findings

VEST is W[2]-hard for parameter k

Computing the k-th homotopy group of certain spaces is W[2]-hard

Results extend complexity classification of topological problems

Abstract

In this short note, we show that the problem of VEST is $W[2]$-hard for parameter $k$. This strengthens a result of Matou\v{s}ek, who showed $W[1]$-hardness of that problem. The consequence of this result is that computing the $k$-th homotopy group of a $d$-dimensional space for $d > 3$ is $W[2]$-hard for parameter $k$.