In the $k$-median problem, $L$ defines as set of feasible facility locations and $C$ defines a set of client locations in a metric space.
The current best approximation guarantee for the problem is $2.675$ due to Byrka et al. Do we have any better approximation guarantees for the problem when $L=C$? If not, is it possible to show that the same $(1+\frac{2}{e})$ hardness of approximation holds when $L = C$?
I am aware that the $k$-center problem admits a $3$-approximation in the general case. And, that it can be improved to $2$-approximation when $L = C$. Moreover, both these bounds are tight in their respective cases. I am hoping for a similar behavior for the $k$-median problem. Any help is appreciated. Thanks.