Single machine scheduling to minimize total weighted tardiness

Abstract The problem of minimizing the total weighted tardiness in single machine scheduling is a well-known strongly NP-hard problem. In this paper, we present an O( n 2 ) time approximation algorithm for the problem, where n is the number of jobs. We further investigate different sub-models of the problem and obtain interesting properties and results. We begin with the model where the job due dates are affine-linear functions of their processing times and the job tardiness weights are proportional to their processing times. For this model, we classify the NP-hardness of the problem with respect to different affine-linear functions, and provide an O( n log n ) algorithm for each of the polynomially solvable problems. The second model we investigate is one where all job due dates have equal slacks, namely the SLK due-date model. Results for this model include: the problem is NP-hard in the ordinary sense; a pseudo-polynomial algorithm with time complexity O( n 2 P ) is derived, where P is the sum of all of the processing times; and a fully polynomial-time approximation scheme is constructed.

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