Finite Dominating Sets for Network Location Problems

Abstract

A research theme involving location on networks, since its inception, has been the identification of a finite dominating set (FDS), or a finite set of points to which an optimal solution must belong. We attempt to unify and generalize results of this sort. We survey the literature and then prove some theorems that subsume most previous results and that are, at the same time, more general than previous results. The paper is aimed primarily at investigators who wish to know whether an FDS exists for a specific problem.

Multifacility ordered median problems on networks: A further analysis

“…(See Mirchandani and Francis [13], Labbé et al [12], Drezner [4], Puerto [17], Drezner and Hamacher [5] and references therein.) The starting point of this development may be considered the node-dominance result of Hakimi [7] and the extensions by Hooker et al [8], which we will show to be essential. In a series of previous papers, a new type of objective function in location theory was introduced and analyzed (see Nickel and Puerto [15], Puerto and Fernández [18], Rodríguez-Chía et al [20], Francis et al [6], Nickel [14], and Kalcsics et al [10]).…”

“…Recall that an FDS is a set which always contains an optimal solution of the problem. See Hooker et al [8] for further details. Identifying a general FDS for this family of problems has important implications on the development of the same kind of efficient algorithms for all these problems simultaneously.…”

“…Finally, recently and independently, Tamir [23] presented another algorithm for the p-facility k-centrum whose complexity for the discrete problem is O ((min(k, p))kpM 5 ). Note that for the continuous version the number of possible candidates is O(M 3 ) instead of O(M) and, therefore, the overall complexity is at least O ((min(k, p))kpM 8 ). Since the p-centdian complexity is O( pM 6 ), the complexity of the algorithm for the more general ordered pmedian problem is not too large.…”

Fuzzy location problems on networks

“…In the median problem, the function to be minimized is the sum of the distances; in the center problem we want to minimize the maximum of the distances. Therefore, the median problem and the center problem are, respectively, formulated by The ÿrst step in the solution of a location problem where all the points are possible solutions is to ÿnd a ÿnite dominant set for the problem, i.e., a ÿnite set that includes, at least, an optimal solution to the problem (see [9]). From the seminal paper of Hakimi [6], dominant ÿnite sets are known for the median, consisting of the vertices, and for the center problem, that consists of the vertices and local centers, i.e., interior points in equilibrium between two vertices.…”

Solving structured multifacility location problems efficiently

“…Of course to fit our format, the solution set Z must be finite. Hooker, Garfinkel and Chen (1991) studied a large number of continuous network location problems in an effort to identify a finite set of points on the network which would contain the new facility locations in an optimal solution. They called such a set of points for a given problem a. finite domination set Problem 1. m-Median Problem (Hakimi, 1964(Hakimi, ,1965 (Goldman, 1971) (Mirchandani and Odoni, 1979) (Dearing, Francis, and Lowe, 1976;Kolen, 1986;Fernandez-Baca, 1989;Chhajed and Lowe, 1990) Problem 6. m-C enter Problem (Hakimi, 1965) (Hooker, Garfinkel and Chen, 1991).…”