Constant factor Approximation Algorithms for Uniform Hard Capacitated Facility Location Problems: Natural LP is not too bad

In this paper, we give first constant factor approximation for capacitated knapsack median problem (CKM) for hard uniform capacities, violating the budget only by an additive factor of $f{max}$ where $f{max}$ is the maximum cost of a facility opened by the optimal and violating capacities by $(2+\epsilon)$ factor. Natural LP for the problem is known to have an unbounded integrality gap when any one of the two constraints is allowed to be violated by a factor less than $2$. Thus, we present a result which is very close to the best achievable from the natural LP. To the best of our knowledge, the problem has not been studied earlier. For capacitated facility location problem with uniform capacities, a constant factor approximation algorithm is presented violating the capacities a little ($1 + \epsilon$). Though constant factor results are known for the problem without violating the capacities, the result is interesting as it is obtained by rounding the solution to the natural LP, which is known to have an unbounded integrality gap without violating the capacities. Thus, we achieve the best possible from the natural LP for the problem. The result shows that natural LP is not too bad. Finally, we raise some issues with the proofs of the results presented in \cite{capkmByrkaFRS2013} for capacitated $k$-facility location problem (C$k$FLP). \cite{capkmByrkaFRS2013} presents $O(1/\epsilon^2)$ approximation violating the capacities by a factor of $(2 + \epsilon)$ using dependent rounding. We first fix these issues using our techniques. Also, it can be argued that (deterministic) pipage rounding cannot be used to open the facilities instead of dependent rounding. Our techniques for CKM provide a constant factor approximation for CkFLP violating the capacities by $(2 + \epsilon)$.