No abstract

No abstract

An r-gathering of customers C to facilities F is an assignment A of C to open facilities F ′ ⊂ F such that r (≥ 2) or more customers are assigned to each open facility. (Each facility needs enough number of customers for its opening.) Then the r-gathering problem finds an r-gathering minimizing a designated cost. Armon gave a simple 3approximation algorithm for the r-gathering problem and proved that with assumption P N P the problem cannot be approximated within a factor of less than 3 for any r ≥ 3. The running time of the 3-approximation algorithm is O(|C | |F | + r |C | + |C | log |C |)). In this paper we improve the running time of the algorithm by (1) removing the sort in the algorithm and (2) designing a simple but efficient data structure.

SUMMARYIn this paper we study a recently proposed variant of the facility location problem, called the r-gathering problem. Given an integer r, a set C of customers, a set F of facilities, and a connecting cost co(c, f ) for each pair of c ∈ C and f ∈ F, an r-gathering of customers C to facilities F is an assignment A of C to open facilities F ⊆ F such that at least r customers are assigned to each open facility. We give an algorithm to find an r-gathering with the minimum cost, where the cost is max c∈C {co(c, A(c))}, when all C and F are on the real line.