The Heterogeneous Capacitated k-Center Problem
In this paper we initiate the study of the heterogeneous capacitated k -center problem: we are given a metric space \(X = (F \cup C, d)\), and a collection of capacities. The goal is to open each capacity at a unique facility location in F, and also to assign clients to facilities so that the number of clients assigned to any facility is at most the capacity installed; the objective is then to minimize the maximum distance between a client and its assigned facility. If all the capacities \(c_i\)’s are identical, the problem becomes the well-studied uniform capacitated k -center problem for which constant-factor approximations are known [7, 22]. The additional choice of determining which capacity should be installed in which location makes our problem considerably different from this problem and the non-uniform generalizations studied thus far in literature. In fact, one of our contributions is in relating the heterogeneous problem to special-cases of the classical santa-claus problem. Using this connection, and by designing new algorithms for these special cases, we get the following results for Heterogeneous Cap-\(k\)-Center.
A quasi-polynomial time \(O(\log n/\epsilon )\)-approximation where every capacity is violated by \((1+\epsilon )\) factor.
A polynomial time O(1)-approximation where every capacity is violated by an \(O(\log n)\) factor.
We get improved results for the soft-capacities version where we can place multiple facilities in the same location.
KeywordsFacility Location Linear Programming Relaxation Cardinality Constraint Capacitate Facility Location Problem Center Instance
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