### 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 $x\in C$, Carath\'eodory'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 \bigO{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 Gr{\"o}tschel, Lov{\'a}sz, and Schrijver applied to one of these subpolytopes.

Original language | Undefined |
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Place of Publication | Enschede |

Publisher | Centre for Telematics and Information Technology (CTIT) |

Number of pages | 14 |

Publication status | Published - 4 Dec 2013 |

### Publication series

Name | CTIT Technical Report Series |
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Publisher | University of Twente, Centre for Telematics and Information Technology |

No. | TR-CTIT-13-25 |

ISSN (Print) | 1381-3625 |

### Keywords

- METIS-300180
- IR-88292
- EWI-24630
- Polyhedral subdivision
- Algorithms
- Zonotopes
- Scheduling polytope
- Convex combination
- Decomposition
- Polyhedral theory

## Cite this

Hoeksma, R. P., Manthey, B., & Uetz, M. J. (2013).

*Decomposition algorithm for the single machine scheduling polytope*. (CTIT Technical Report Series; No. TR-CTIT-13-25). Enschede: Centre for Telematics and Information Technology (CTIT).