My question is whether there exist algorithms to determine a minimal (or close to minimal) cover of a Riemannian manifold (or some subset thereof) with balls of a fixed radius r>0.



Algorithms for finding minimal ball covers
As it is a version of set-cover, the problem is NP-hard.
The following is a special case:



> 
> **(1)** Chepoi, Victor, and Bertrand Estellon. "Packing and covering $\delta$-hyperbolic spaces by balls." In *Approximation, Randomization, and Combinatorial Optimization.* Algorithms and Techniques, pp. 59-73. Springer, Berlin, Heidelberg, 2007.
>  [Springer link](https://link.springer.com/chapter/10.1007/978-3-540-74208-1_5).
>    
> 
>  "In this note, we consider the problem of covering and packing by balls and
>  union of balls of hyperbolic metric spaces and graphs."
> 
> 
> 




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> 
> **(2)** Singhof, Wilhelm. "Minimal coverings of manifolds with balls." *Manuscripta Mathematica* 29, no. 2-4 (1979): 385-415.
>  [Springer link](https://link.springer.com/article/10.1007%2FBF01303636).
> 
> 
> 


My understanding is that Singhof links the covering number $B(M)$ of a manifold $M$
to the Lusternik-Schnirelmann category $\operatorname{cat} M$ of a manifold.


You might find later literature, especially for (2), that relies on or 
expands upon these results. 


