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

Artur Czumaj, Christiane Lammersen, Morteza Monemizadeh, Christian Sohler

Abstract: We consider the Euclidean facility location problem with uniform opening cost. In this problem, we are given a set of n points P sube ℝ2 and an opening cost f ∊ ℝ+, and we want to find a set of facilities F ⊆ ℝ2 that minimizes 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 which is (n log2 n log log n) for any 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, …, Δ}2. The streaming algorithm uses 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 for each Pi one needs to open only a small, polylogarithmic number of facilities), Σi OPT(Pi) ≤ (1 + ε) · OPT(P), where for a point set P, OPT(P) denotes the cost of an optimal solution for P. The decomposition 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.