  Capacitated k-median is one of the few outstanding optimization problems for
which the existence of a polynomial time constant factor approximation
algorithm remains an open problem. In a series of recent papers algorithms
producing solutions violating either the number of facilities or the capacity
by a multiplicative factor were obtained. However, to produce solutions without
violations appears to be hard and potentially requires different algorithmic
techniques. Notably, if parameterized by the number of facilities $k$, the
problem is also $W[2]$ hard, making the existence of an exact FPT algorithm
unlikely. In this work we provide an FPT-time constant factor approximation
algorithm preserving both cardinality and capacity of the facilities. The
algorithm runs in time $2^{\mathcal{O}(k\log k)}n^{\mathcal{O}(1)}$ and
achieves an approximation ratio of $7+\varepsilon$.
