(1+ Є)-approximation for facility location in data streams

We consider the Euclidean facility location problem with uniform opening cost. In this problem, we are given a set of n points P ⊆ R and an opening cost f ∈ R, and we want to find a set of facilities F ⊆ R that minimizes f · |F |+ ∑ p∈P min q∈F d(p, q) , where d(p, q) is the Euclidean distance between p and q. We obtain two main results: • A (1 + ε)-approximation algorithm with running time O(n log n log log n) for constant ε, • The first (1 + ε)-approximation algorithm for the cost of the facility location problem for dynamic geometric data streams, i.e., when the stream consists of insert and delete operations of points from a discrete space {1, . . . ,∆}. The streaming algorithm uses ( log ∆ ε )O(1) space. Our PTAS is significantly faster than any previously known (1 + ε)-approximation algorithm for the problem, and is also relatively simple. Our algorithm for dynamic geometric data streams is the first (1 + ε)-approximation algorithm for the cost of the facility location problem with polylogarithmic space, and it resolves an open problem in the streaming area. Both algorithms are based on a novel and simple decomposition of an input point set P into small subsets Pi, such that: • the cost of solving the facility location problem for each Pi is small (which means that one needs to open only a small, polylogarithmic number of facilities), • ∑ i OPT(Pi) ≤ (1 + ε) ·OPT(P ), where for a point set P , OPT(P ) denotes the cost of an optimal solution for P . ∗Research partially supported by the EU within the 7th Framework Programme under contract No. 255827, an Ebco Eppich, a PIMS Fellowship, the Centre for Discrete Mathematics and its Applications (DIMAP), and by EPSRC awards EP/D063191/1 and EP/G064679/1. Part of the work on this paper has been supported by Deutsche Forschungsgemeinschaft (DFG) within the Collaborative Research Center SFB 876 “Providing Information by Resource-Constrained Analysis”, project A2. †Department of Computer Science and Centre for Discrete Mathematics and its Applications (DIMAP), University of Warwick, [email protected]. ‡School of Computing Science, Simon Fraser University, [email protected]. §Institute for Computer Science, University of Frankfurt, [email protected]. Supported in part by BMBF grant 06FY9097. ¶Department of Computer Science, Technische Universität Dortmund, [email protected]. Part of this work has been supported by Deutsche Forschungsgemeinschaft (DFG) within the Collaborative Research Center SFB 876 ”Providing Information by ResourceConstrained Analysis”, project A2. The partitioning can be used directly to obtain the PTAS by splitting the point set in the subsets and efficiently solve the problem for each subset independently. By combining our partitioning with techniques to process dynamic data streams of sampling from the cells of the partition and estimating the cost from the sample, we obtain our data streaming algorithm.