Approximating the throughput of multiple machines under real-time scheduling

Abstract

We consider the following fundamental scheduling problem. The input to the problem consists of n jobs and k machines. Each of the jobs is associated with a release time, a deadline, a weight, anda processing time on each of the machines. The goal is to finda nonpreemptive sched ule that maximizes the weight of jobs that meet their respective deadlines. We give constant factor approximation algorithms for four variants of the problem, depending on the type of the machines (identical vs. unrelated) and the weight of the jobs (identical vs. arbitrary). All these variants are known to be NP-hard , andthe two variants involving unrelatedmachines are also MAX-SNP hard . The specific results obtainedare as follows: • For id entical job weights andunrelatedmachines: a greed y 2-approximation algorithm. • For identical job weights and k identical machines: the same greedy algorithm achieves a tight (1+1/k) k (1+1/k)k−1 approximation factor. • For arbitrary job weights anda single machine: an LP formulation achieves a 2-approximation for polynomially bound edintegral input anda 3-approximation for arbitrary input. For unrelatedmachines, the factors are 3 and4, respectively. • For arbitrary job weights and k id entical machines: the LP-basedalgorithm appliedre- peatedly achieves a (1+1/k) k (1+1/k)k−1 approximation factor for polynomially bounded integral input anda (1+1/2k) k (1+1/2k)k−1 approximation factor for arbitrary input. • For arbitrary job weights andunrelatedmachines: a combinatorial (3 + 2 √ 2 ≈ 5.828)- approximation algorithm.