ISCO 2014: Combinatorial Optimization pp 280-291

# Decomposition Algorithm for the Single Machine Scheduling Polytope

• Ruben Hoeksma
• Bodo Manthey
• Marc Uetz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8596)

## Abstract

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.

## Keywords

Completion Time Convex Hull Convex Combination Idle Time Single Machine
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Notes

### Acknowledgements

We thank Maurice Queyranne for pointing us to the paper by Yasutake et al. [16], and Marc Pfetsch and Michael Joswig for helpful remarks concerning zonotopes.

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