The Directional p-Median Problem with Applications to Traffic Quantization and Multiprocessor Scheduling

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 thesis, we treat the p-median problem under a new distance measure, the directional rectilinear distance, which requires the nearest supply point for a given demand point to lie above and to the right of it. This restriction has applications to multiprocessor scheduling of periodic tasks as well as to traffic quantization and Quality of Service scheduling in packet-switched computer networks. We show that the directional p-median problem is polynomially solvable in one dimension and give two algorithms. We prove the problem NP-hard in two or more dimensions and then present an efficient heuristic to solve it. Compared to the robust Teitz and Bart heuristic for p-median, our heuristic enjoys substantial speedup while sacrificing little in terms of solution quality, making it an ideal choice for our target applications with thousands of demand points.