By Kai-Uwe Sattler

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4. J. R. Correa and A. S. Schulz. Single machine scheduling with precedence constraints. Mathematics of Operations Research, 30(4):1005–1021, 2005. Extended abstract in Proceedings of the 10th Conference on Integer Programming and Combinatorial Optimization (IPCO 2004), pages 283–297. 5. B. Dushnik and E. Miller. Partially ordered sets. American Journal of Mathematics, 63:600–610, 1941. 6. S. Felsner and R. M¨ ohring. Semi order dimension two is a comparability invariant. Order, (15):385–390, 1998.

Call such interval vectors candidate interval vectors. We show that there exists a certain “minimal” set of independent interval vectors R2 , such that any feasible collection of candidate interval vectors can be transformed (without any loss) into one that only uses vectors from R2 . Since R2 only contains independent vectors, |R2 | ≤ z/2. Now, if Opt uses close to z/2 vectors in its solution, then it must have discarded very few vectors from R2 . This allows us to use the known results for the generalized caching problem considered by [2, 3], and hence ﬁnd a solution where the number of discarded intervals is no more than a constant times the number of intervals discarded by Opt.

We call a solution that consists only of interval vectors that begin at upticks and end at downticks a canonical solution. We now show how to view any canonical solution to the setup problem as a graph, which will allow us to characterize the value of an optimum solution exactly. Let Iu denote the index set of the upticks and Id denote the index set of the downticks. Consider a canonical solution S = {V1 , . . , Vt }, where Vi = ( i , ri , bi ) are interval vectors. Since all interval vectors begin at upticks and end at downticks if some position j is an uptick, then the sum of heights of all interval vectors that begin at position j is exactly equal to ΔA (j).