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

Abstract. In this paper, we study the uniform hard capacitated k facility location problem (CkFLP) and knapsack median problem (CKM). Natural LP of both the problems have an unbounded integrality gap. Byrka et al. in [5] present an (O(1/ǫ)) for CkFLP violating cpapcities by a factor of (2 + ǫ). However, the proofs in [5] do not seem to work. In this paper, we first raise the issues in [5] and then provide a constant factor approximation solution violating the capacities by 3 and opening at most k + 1 facilities. We also give first constant factor approximation for CKM violating the budget only by an additive factor of fmax where fmax is the maximum cost of a facility opened by the optimal and violating capacities by 3 factor. To the best of our knowledge, no constant factor approximation is known for the problem even with capacity/budget/both violations. For capacitated facility location problem with uniform capacities, a constant factor approximation algorithm is presented violating the capacities a little (1 + ǫ). 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.