Computational complexity of topological invariants

We answer the following question posed by Lechuga: Given a simply-connected space $X$ with both $H_{\ast }(X,\text{qq})$ and ${\pi }_{\ast }(X)\otimes \text{qq}$ 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 $k\ge 3$) we show that (even stricter versions of the) problems above are $\mathbf{N}\mathbf{P}$-hard.