Approximating Single Machine Scheduling with Scenarios

We consider the single machine scheduling problem to minimize the weighted sum of completion times under precedence constrains. In a series of recent papers it was established that this scheduling problem is a special case of minimum weighted vertex cover. In this paper we show that the vertex cover graph associated with the scheduling problem is exactly the graph of incomparable pairs defined in dimension theory of partial orders. Exploiting this relationship allows us to present a framework for obtaining (2 − 2/f)-approximation algorithms provided that the set of precedence constraints has fractional dimension at most f. Our approach yields the best known approximation ratios for all previously considered special classes of precedence constraints, and it provides the first results for bounded degree and orders of interval dimension 2. On the negative side, we show that the addressed problem remains NP-hard even when restricted to the special case of interval orders. Furthermore, we prove that the general problem, if a fixed cost present in all feasible schedules is ignored, becomes as hard to approximate as vertex cover. We conclude by giving the first inapproximability result for this problem, showing under a widely believed assumption that it does not admit a polynomial time approximation scheme.1 Key words: single-machine scheduling; vertex cover; fractional dimensio