On polynomial solvability of the high multiplicity total weighted tardiness problem

In a recent paper Hochbaum et al. developed a polynomial algorithm for solving a scheduling problem of minimizing the total weighted tardiness for a large number of unit length jobs which can be partitioned into a few sets of jobs with identical due dates and penalty weights. The number of unit jobs in a set is called the multiplicity of that set. The problem was formulated in Hochbaum et al. as an integer quadratic nonseparable transportation problem and solved, in polynomial time, independent of the size of the multiplicities and the due dates but depending on the penalty weights. In this paper the authors show how to solve the above problem in polynomial time which is independent of the sizes of the weights. The running time of the algorithm depends on the dimension of the problem and only the size of the maximal difference between two consecutive due dates. In the case where the due dates are large, but the size of the maximal difference between two consecutive due dates is polynomially bounded by the dimension of the problem, the algorithm runs in strongly polynomial time.