The directional p-median problem: Definition, complexity, and algorithms

Abstract

An instance of a p-median problem gives n demand points. The objective is to locate p supply points in order to minimize the total distance of the demand points to their nearest supply point. p-Median is polynomially solvable in one dimension but NP-hard in two or more dimensions, when either the Euclidean or the rectilinear distance measure is used. In this paper, we treat the p-median problem under a new distance measure, the directional rectilinear distance, which requires the assigned supply point for a given demand point to lie above and to the right of it. In a previous work, we showed that the directional p-median problem is polynomially solvable in one dimension; we give here an improved solution through reformulating the problem as a special case of the constrained shortest path problem. We have previously proven that the problem is NP-complete in two or more dimensions; we present here an efficient heuristic to solve it. Compared to the robust Teitz and Bart heuristic, our heuristic enjoys substantial speedup while sacrificing little in terms of solution quality, making it an ideal choice for real-world applications with thousands of demand points.

On Bandwidth Tiered Service

“…We present a new dynamic programming algorithm for determining the optimal service tiers whose complexity is linear in the number of users, and hence scales to very large numbers of users (a typical scenario); this represents a significant improvement over the complexity of the dynamic programming algorithm of [28] that is quadratic in the number of users. The new algorithm is based on a novel dynamic programming formulation that exploits intrinsic properties of the problem.…”

“…The new algorithm is based on a novel dynamic programming formulation that exploits intrinsic properties of the problem. We also consider a new version of the service tier selection problem in which the objective is to optimize jointly the number of tiers and the magnitude of each tier; in contrast, the number of tiers is provided as an input parameter in the problem we considered in [28]. This problem is of practical significance since typically there is a cost incurred for each additional service tier offered.…”

“…Recently [28], we developed a theoretical framework based on location theory for selecting the service tiers optimally, eliminating the need for guesswork and ad hoc approaches. Our work was based on the observation that, by mapping a user to the next higher offered service level, a tiered-service network may use more resources that a continuous-rate one to satisfy the same set of requests for service; alternatively, for a given amount of resources, a tiered-service network will be able to accommodate a smaller fraction of user requests than a continuous-rate one.…”

TDM Emulation in Packet-Switched Networks

“…Integer K corresponds to the largest possible multiple of r , i.e., K = xn r , where x n is the largest demand. This version of STO is a special case of the directional pmedian problem we defined and studied in [4]. Consequently, this problem can be solved in time O(pK) using a dynamic programming algorithm; the reader is referred to [4] for details on the dynamic programming formulation.…”

Using dual feasible functions to construct fast lower bounds for routing and location problems

“…Besides location and distribution applications, p-Median problems can also arise in rather unexpected fields such as cluster analysis (see references in the introduction of [1]), processor scheduling or packet management in networks [14].…”