Faster Min-Max <i>r</i>-Gatherings

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.

Assigning Proximity Facilities for Gatherings

“…If C and F are sets of points in a metric space then Armon [3] gave a simple O(|C|(|F |+r + log|C|)) time 3approximation algorithm for the 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 slightly improved [8].…”