We consider the scheduling problem of minimizing the average weighted job completion time on a single machine under precedence constraints. We show that this problem with arbitrary job weights, the special case of the problem where all job weights are one, and several other special cases of the problem all have the same approximability threshold with respect to polynomial time approximation algorithms. Moreover, for the special case of interval order precedence constraints and for the special case of convex bipartite precedence constraints, we give a polynomial time approximation algorithm with worst case performance guarantee arbitrarily close to the golden ratio 1/2(1 + 5~(1/2)) ≈ 1.61803.
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