We study the capacitated k-facility location problem, in which we are given a set of clients with demands, a set of facilities with capacities and a positive integer k. It costs fi to open facility i, and cij for facility i to serve one unit of demand from client j. The objective is to open at most k facilities serving all the demands and satisfying the capacity constraints while minimizing the sum of service and opening costs. In this paper, we give the first fully polynomial time approximation scheme (FPTAS) for the single-sink (single-client) capacitated k-facility location problem. Then, we show that the capacitated k-facility location problem with uniform capacities is solvable in polynomial time if the number of clients is fixed by reducing it to a collection of transportation problems. Third, we analyze the structure of extreme point solutions, and examine the efficiency of this structure in designing approximation algorithms for capacitated k-facility location problems. Finally, we extend our results to obtain an improved approximation algorithm for the capacitated facility location problem with uniform opening costs.@en

We study the k-level uncapacitated facility location problem (k-level UFL) in which clients need to be connected with paths crossing open facilities of k types (levels). In this paper we first propose an approximation algorithm that for any constant k, in polynomial time, delivers solutions of cost at most αk times OPT, where αk is an increasing function of k, with limk→∞αk=3. Our algorithm rounds a fractional solution to an extended LP formulation of the problem. The rounding builds upon the technique of iteratively rounding fractional solutions on trees (Garg, Konjevod, and Ravi SODA’98) originally used for the group Steiner tree problem. We improve the approximation ratio for k-level UFL for all k ≥ 3, in particular we obtain the ratio equal 2.02, 2.14, and 2.24 for k = 3,4, and 5. Second, we give a simple interpretation of the randomization process (Li ICALP’2011) for 1-level UFL in terms of solving an auxiliary (factor revealing) LP. Armed with this simple view point, we exercise the randomization on our algorithm for the k-level UFL. We further improve the approximation ratio for all k ≥ 3, obtaining 1.97, 2.09, and 2.19 for k = 3,4, and 5. Third, we extend our algorithm to the k-level UFL with penalties (k-level UFLWP), in which the setting is the same as k-level UFL except that the planner has the option to pay a penalty instead of connecting chosen clients.@en