What is the name of this problem (the dual of the asymmetric k-center problem)
It is not entirely clear to me where the warehouses are allowed to be placed. I will assume here that a warehouse can be placed in any city given by the input1.

instead of minimizing the distance [$t$], I want to fix it and minimizing the number of warehouses. is this problem NP-Complete also?

Observe that we may assume the maximum allowed distance $t=1$, because we can simply scale the input by a factor of $1/t$. This problem is known as the DISCRETE UNIT DISK COVER problem (DUDC).
DUDC is NP-hard. To see why, first note that the problem is equivalent to the dominating set problem on unit disk graphs.2 The dominating set problem is NP-hard for general graphs, but this does not imply that it remains hard for unit disk graphs in particular. However, the dominating set problem is known to be NP-hard on &quot;grid graphs&quot;: unit disk graphs that have an embedding where all vertices lie on an integer grid with disks of radius $1/2$.3 Since all grid graphs are unit disk graphs, the dominating set problem is NP-hard for unit disk graphs, and hence DUDC is NP-hard.

1: Another option is that there is a finite number of fixed sites for warehouses, which are not necessarily inside a city. But this has the problem I discuss here as a special case, so my discussion here shows that both problems are in fact NP-hard.
2: Technically, we need to restrict ourselves to unit disk graphs for which we can compute an embedding in polynomial time, as this is not known to be possible for an arbitrary unit disk graph. Fortunately, this is possible for &quot;grid graphs&quot; (what helps is that these graphs are planar, unlike general unit disk graphs), so the construction works out fine.
3: See the paper Unit disk graphs by Clark, Colbourn and Johnson. They also note there exists an earlier proof for the hardness of the DUDC problem in the conference paper &quot;S. Masuyama, T. Ibaraki and T. Hasegawa, The computational complexity of the M-center
problems in the plane&quot; from 1981, but this paper appears to be accessible only for members of the Japanese institute that published the conference proceedings)