Computational complexity of topological invariants

TL;DR

This paper proves that computing topological invariants like cup-length and rational LS-category for certain spaces is NP-hard, by reducing from the graph k-colorability problem, highlighting their computational difficulty.

Contribution

It establishes the NP-hardness of computing key topological invariants for simply-connected spaces with finite-dimensional rational homology and homotopy groups.

Findings

Computing cup-length is NP-hard.

Computing rational LS-category is NP-hard.

Reduces from graph k-colorability problem.

Abstract

We answer the following question posed by Lechuga: Given a simply-connected space $X$ with both $H_*(X,\qq)$ and $\pi_*(X)\otimes \qq$ being finite-dimensional, what is the computational complexity of an algorithm computing the cup-length and the rational Lusternik--Schnirelmann category of $X$? Basically, by a reduction from the decision problem whether a given graph is $k$-colourable (for $k\geq 3$) we show that (even stricter versions of the) problems above are $\mathbf{NP}$-hard.