(1 + ε)-Approximation for Facility Location in Data Streams∗
Artur Czumaj, Christiane Lammersen, Morteza Monemizadeh, Christian Sohler · 2015
We consider the Euclidean facility location problem with uni-form opening cost. In this problem, we are given a set of n points P ⊆ R2 and an opening cost f ∈ R+, and we want to find a set of facilities F ⊆ R2 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 log2 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,...,∆}2. 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 decompo-sition 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