Algorithmica (1999) 24: 29–49 Algorithmica © 1999 Springer-Verlag New York Inc. On-Line Resource Management with Application to Routing and Scheduling 1 S. Leonardi 2 and A. Marchetti-Spaccamela 2 Abstract. We propose a framework to model on-line resource management problems based on an on-line version of positive linear programming. We consider both min cost problems and max benefit problems and propose logarithmic competitive algorithms that are optimal up to a constant factor. The proposed framework provides a general methodology that applies to a wide class of on-line problems: shop scheduling, packet routing, and in general a class of packing and assignment problems. Previously studied problems as on-line multiprocessor scheduling and on-line virtual circuit routing can also be modeled within this framework. Key Words. On-line algorithms, Competitive analysis, Routing, Scheduling, Linear programming. 1. Introduction. In this paper we consider on-line resource management problems, on-line optimization problems concerned with the efficient use of resources to accom- plish a set of tasks—also called jobs in the following. Informally, an on-line resource management problem is defined by a sequence of jobs that are presented one at a time. The processing of a job can be done following one of a set of possible alternatives. Different alternatives make use of different sets of resources. Decisions about how to process a job must be taken when the job is presented, without knowledge of the jobs that will follow in the sequence. This class of problems includes routing, scheduling, and financial problems. The application fields we consider give rise to both cost problems and benefit problems. In a cost problem every job can be accomplished in several ways. The goal of the on-line algorithm is to minimize the amount of resources necessary to deal with all the jobs in a sequence. In a benefit problem the amount of resources available is fixed at the beginning and a revenue is obtained for each accepted job. The goal is to maximize the benefit gained over a sequence of jobs without exceeding the available resources. Within this goal, a job is possibly not served at all. We present a framework based on an on-line version of positive linear programming to model on-line resource management problems. Competitive Analysis. We use competitive analysis [15], [16], [20], [19], [24], [30] for the analysis of on-line algorithms; competitive analysis compares an on-line 1 A preliminary version of this paper appeared in the Proceedings of the 22nd International Colloquium on Automata, Languages, and Programming, LNCS 944, Springer-Verlag, 1995. This work was partly supported by EC ESPRIT BRA Alcom II under Contract No. 7141, EC HCM Project Map, and by Italian Ministry of Scientific Research Project 40% “Efficienza di Algoritmi e Progetto di Strutture Informative.” 2 Dipartimento di Informatica e Sistemistica, Universit` a di Roma “La Sapienza,” via Salaria 113, 00198-Roma, Italy. {leon,marchetti}@dis.uniroma1.it. Received December 18, 1996; revised March 2, 1997. Communicated by A. B. Borodin.