### Abstract

The computational complexity of the partition problem, which concerns the partitioning of a set of n vectors in d-space into p parts so as to maximize an objective function which is convex on the sum of vectors in each part, is determined by the number of vertices of the corresponding p-partition polytope defined to be the convex hull in (d × p)-space of all solutions. In this article, introducing and using the class of Momentopes, we establish the lower bound v_{p,d}(n) = ω(n^{[(d-1)/2]p}) on the maximum number of vertices of any p-partition polytope of a set of n points in d-space, which is quite compatible with the recent upper bound v_{p,d}(n) = O(n^{d(p-1)-1}), implying the same bound on the complexity of the partition problem. We also discuss related problems on the realizability of Davenport-Schinzel sequences and describe some further properties of Momentopes.

Original language | English (US) |
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Pages (from-to) | 409-417 |

Number of pages | 9 |

Journal | Discrete and Computational Geometry |

Volume | 27 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2002 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics