Advanced Approximation Algorithms ( CMU 18 - 854 B , Spring 2008 ) Lecture 6 : Facility location : greedy and local search algorithms

Recall the greedy algorithm for non-metric facility location that was done in Lecture 4, which was obtained by formulating the problem as set cover. In this section we present a modified greedy algorithm for the metric facility location problem that achieves a constant approximation ratio. Let the facility location instance consist of clients D, facilities F , metric d on D ∪ F , and facility opening costs fi ∈ R+ for each i ∈ F . Then the goal is to open a set F ∗ ⊆ F of facilities that minimizes ∑ i∈F ∗ fi + ∑ j∈D d(j, F ∗). Above for any client j ∈ D, d(j, F ∗) = mini∈F ∗ d(j, i) denotes the distance from j to its nearest facility in F ∗. As seen in Lecture 4, the facility location problem can be cast as a set covering problem where elements are the clients D, and sets correspond to ’stars’ centered at some facility. Any set can be represented as (i, A) where i ∈ F is a facility and A ⊆ D is a subset of clients; the cost of this set is fi + ∑ j∈A d(i, j). The modified greedy algorithm is as follows. • Set F ′ ← φ.