### Abstract

The partition problem concerns the partitioning of n given 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. The problem has applications in diverse fields that include clustering, inventory, scheduling and statistical hypothesis testing. Since the objective function is convex, the partition problem can be reduced to the problem of maximizing the same objective over the partition polytope, defined to be the convex hull of all solutions. In this article we completely characterize the vertices of partition polytopes when either d=2 or p=2, and determine the maximum number of vertices of any partition polytope of n vectors when d=2 or p=2 up to a constant factor. Our characterization implies a bijection between vertices of the polytope of 0-separable partitions, and is best possible in the sense that there are examples in the literature showing the analogue fails already for p=d=3. Our enumerative results provide lower bounds on the time complexity needed to solve the partition problem when the objective function is presented by an oracle.

Original language | English (US) |
---|---|

Pages (from-to) | 1-15 |

Number of pages | 15 |

Journal | Discrete Applied Mathematics |

Volume | 124 |

Issue number | 1-3 |

DOIs | |

State | Published - Dec 15 2002 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete Applied Mathematics*,

*124*(1-3), 1-15. https://doi.org/10.1016/S0166-218X(01)00326-2

**Vertex characterization of partition polytopes of bipartitions and of planar point sets.** / Aviran, Sharon; Lev-Tov, Nissan; Onn, Shmuel; Rothblum, Uriel G.

Research output: Contribution to journal › Article

*Discrete Applied Mathematics*, vol. 124, no. 1-3, pp. 1-15. https://doi.org/10.1016/S0166-218X(01)00326-2

}

TY - JOUR

T1 - Vertex characterization of partition polytopes of bipartitions and of planar point sets

AU - Aviran, Sharon

AU - Lev-Tov, Nissan

AU - Onn, Shmuel

AU - Rothblum, Uriel G.

PY - 2002/12/15

Y1 - 2002/12/15

N2 - The partition problem concerns the partitioning of n given 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. The problem has applications in diverse fields that include clustering, inventory, scheduling and statistical hypothesis testing. Since the objective function is convex, the partition problem can be reduced to the problem of maximizing the same objective over the partition polytope, defined to be the convex hull of all solutions. In this article we completely characterize the vertices of partition polytopes when either d=2 or p=2, and determine the maximum number of vertices of any partition polytope of n vectors when d=2 or p=2 up to a constant factor. Our characterization implies a bijection between vertices of the polytope of 0-separable partitions, and is best possible in the sense that there are examples in the literature showing the analogue fails already for p=d=3. Our enumerative results provide lower bounds on the time complexity needed to solve the partition problem when the objective function is presented by an oracle.

AB - The partition problem concerns the partitioning of n given 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. The problem has applications in diverse fields that include clustering, inventory, scheduling and statistical hypothesis testing. Since the objective function is convex, the partition problem can be reduced to the problem of maximizing the same objective over the partition polytope, defined to be the convex hull of all solutions. In this article we completely characterize the vertices of partition polytopes when either d=2 or p=2, and determine the maximum number of vertices of any partition polytope of n vectors when d=2 or p=2 up to a constant factor. Our characterization implies a bijection between vertices of the polytope of 0-separable partitions, and is best possible in the sense that there are examples in the literature showing the analogue fails already for p=d=3. Our enumerative results provide lower bounds on the time complexity needed to solve the partition problem when the objective function is presented by an oracle.

UR - http://www.scopus.com/inward/record.url?scp=84867955940&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867955940&partnerID=8YFLogxK

U2 - 10.1016/S0166-218X(01)00326-2

DO - 10.1016/S0166-218X(01)00326-2

M3 - Article

AN - SCOPUS:84867955940

VL - 124

SP - 1

EP - 15

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 1-3

ER -