Assigning Proximity Facilities for Gatherings

Abstract

In this paper we study a recently proposed variant of the r-gathering problem. An r-gathering of customers C to facilities F is an assignment A of C to open facilities F ′ ⊂ F such that r or more customers are assigned to each open facility. (Each facility needs enough number of customers to open.) Given an opening cost op( f ) for each f ∈ F, and a connecting cost co(c, f ) for each pair of c ∈ C and f ∈ F, the cost of an r-gathering A is max{max c∈C {co(c, A(c))}, max f ∈F ′ {op( f )} }. The r-gathering problem consists of finding an r-gathering having the minimum cost. Assume that F is a set of locations for emergency shelters, op( f ) is the time needed to prepare a shelter f ∈ F, and co(c, f ) is the time needed for a person c ∈ C to reach assigned shelter f = A(c) ∈ F. Then an r-gathering corresponds to an evacuation plan such that each open shelter serves r or more people, and the r-gathering problem consists of finding an evacuation plan minimizing the evacuation time span. However in a solution above some person may be assigned to a farther open shelter although it has a closer open shelter. It may be difficult for the person to accept such an assignment for an emergency situation. Therefore, Armon considered the problem with one more additional constraint, that is, each customer should be assigned to a closest open facility, and gave a 9-approximation polynomial-time algorithm for the problem. We have designed a simple 3approximation algorithm for the problem. The running time is O(r |C | |F |).