In this paper we introduce two general techniques for the design and analysis of approximation algorithms for 𝒩𝒫-hard scheduling problems in which the objective is to minimize the weighted sum of the job completion times. For a variety of scheduling models, these techniques yield the first algorithms that are guaranteed to find schedules that have objective function value within a constant factor of the optimum. In the first approach, we use an optimal solution to a linear programming relaxation in order to guide a simple list-scheduling rule. Consequently, we also obtain results about the strength of the relaxation. Our second approach yields on-line algorithms for these problems: in this setting, we are scheduling jobs that continually arrive to be processed and, for each time t, we must construct the schedule until time t without any knowledge of the jobs that will arrive afterwards. Our on-line technique yields constant performance guarantees for a variety of scheduling environments, and in some cases essentially matches the performance of our off-line LP-based algorithms.

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We consider the problem of finding near-optimal solutions for a variety of NP-hard scheduling problems for which the objective is to minimize the total weighted completion time. Recent work has led to the development of several techniques that yield constant worst-case bounds in a number of settings. We continue this line of research by providing improved performance guarantees for several of the most basic scheduling models, and by giving the first constant performance guarantee for a number of more realistically constrained scheduling problems. For example, we give an improved performance guarantee for minimizing the total weighted completion time subject to release dates on a single machine, and subject to release dates and/or precedence constraints on identical parallel machines. We also give improved bounds on the power of preemption in scheduling jobs with release dates on parallel machines.We give improved on-line algorithms for many more realistic scheduling models, including environments with parallelizable jobs, jobs contending for shared resources, tree precedence-constrained jobs, as well as shop scheduling models. In several of these cases, we give the first constant performance guarantee achieved on-line. Finally, one of the consequences of our work is the surprising structural property that there are schedules that simultaneously approximate the optimal makespan and the optimal weighted completion time to within small constants. Not only do such schedules exist, but we can find approximations to them with an on-line algorithm.