Decomposition Algorithm for the Single Machine Scheduling Polytope
Given an \(n\)-vector \(p\) of processing times of jobs, the single machine scheduling polytope \(C\) arises as the convex hull of completion times of jobs when these are scheduled without idle time on a single machine. Given a point \(x\in C\), Carathéodory’s theorem implies that \(x\) can be written as convex combination of at most \(n\) vertices of \(C\). We show that this convex combination can be computed from \(x\) and \(p\) in time \(O (n^2)\), which is linear in the naive encoding of the output. We obtain this result using essentially two ingredients. First, we build on the fact that the scheduling polytope is a zonotope. Therefore, all of its faces are centrally symmetric. Second, instead of \(C\), we consider the polytope \(Q\) of half times and its barycentric subdivision. We show that the subpolytopes of this barycentric subdivison of \(Q\) have a simple, linear description. The final decomposition algorithm is in fact an implementation of an algorithm proposed by Grötschel, Lovász, and Schrijver applied to one of these subpolytopes.
KeywordsCompletion Time Convex Hull Convex Combination Idle Time Single Machine
We thank Maurice Queyranne for pointing us to the paper by Yasutake et al. , and Marc Pfetsch and Michael Joswig for helpful remarks concerning zonotopes.
- 1.Cai, Y., Daskalakis, C., Weinberg, S.M.: Optimal multi-dimensional mechanism design: reducing revenue to welfare maximization. In: Proceedings of 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 130–139. IEEE (2012)Google Scholar
- 4.Fonlupt, J., Skoda, A.: Strongly polynomial algorithm for the intersection of a line with a polymatroid. In: Cook, W., Lovász, L., Vygen, J. (eds.) Research Trends in Combinatorial Optimization, pp. 69–85. Springer, Heidelberg (2009)Google Scholar
- 6.Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization: Algorithms and Combinatorics, vol. 2. Springer, Heidelberg (1988)Google Scholar
- 7.Hoeksma, R., Manthey, B., Uetz, M.: Decomposition algorithm for the single machine scheduling polytope. Technical Report TR-CTIT-13-25, CTIT, University of Twente. http://eprints.eemcs.utwente.nl/24630/
- 10.Lee, C.W.: Subdivisions and triangulations of polytopes. In: Handbook of Discrete and Computational Geometry, chapter 17, 2nd edn. Chapman & Hall/CRC, Beca Raton (2004)Google Scholar
- 14.Queyranne, M., Schulz, A.S.: Polyhedral approaches to machine scheduling. Preprint 408–1994, TU Berlin (1994)Google Scholar