### Abstract

Given two finite sets of points X, Y in ℝ^{n} which can be separated by a nonnegative linear function, and such that the componentwise minimum of any two distinct points in X is dominated by some point in Y, we show that |X| ≤ n|Y|. As a consequence of this result, we obtain quasi-polynomial time algorithms for generating all maximal integer feasible solutions for a given monotone system of separable inequalities, for generating all p-inefficient points of a given discrete probability distribution, and for generating all maximal empty hyper-rectangles for a given set of points in ℝ^{n}. This provides a substantial improvement over previously known exponential algorithms for these generation problems related to Integer and Stochastic Programming, and Data Mining. Furthermore, we give an incremental polynomial time generation algorithm for monotone systems with fixed number of separable inequalities, which, for the very special case of one inequality, implies that for discrete probability distributions with independent coordinates, both p-efficient and p-inefficient points can be separately generated in incremental polynomial time.

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

Pages (from-to) | 543-555 |

Number of pages | 13 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Volume | 2719 |

State | Published - Dec 1 2003 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*,

*2719*, 543-555.

}

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*, vol. 2719, pp. 543-555.

**An intersection inequality for discrete distributions and related generation problems.** / Boros, Endre; Elbassioni, Khaled; Gurvich, Vladimir; Khachiyan, Leonid; Makino, Kazuhisa.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An intersection inequality for discrete distributions and related generation problems

AU - Boros, Endre

AU - Elbassioni, Khaled

AU - Gurvich, Vladimir

AU - Khachiyan, Leonid

AU - Makino, Kazuhisa

PY - 2003/12/1

Y1 - 2003/12/1

N2 - Given two finite sets of points X, Y in ℝn which can be separated by a nonnegative linear function, and such that the componentwise minimum of any two distinct points in X is dominated by some point in Y, we show that |X| ≤ n|Y|. As a consequence of this result, we obtain quasi-polynomial time algorithms for generating all maximal integer feasible solutions for a given monotone system of separable inequalities, for generating all p-inefficient points of a given discrete probability distribution, and for generating all maximal empty hyper-rectangles for a given set of points in ℝn. This provides a substantial improvement over previously known exponential algorithms for these generation problems related to Integer and Stochastic Programming, and Data Mining. Furthermore, we give an incremental polynomial time generation algorithm for monotone systems with fixed number of separable inequalities, which, for the very special case of one inequality, implies that for discrete probability distributions with independent coordinates, both p-efficient and p-inefficient points can be separately generated in incremental polynomial time.

AB - Given two finite sets of points X, Y in ℝn which can be separated by a nonnegative linear function, and such that the componentwise minimum of any two distinct points in X is dominated by some point in Y, we show that |X| ≤ n|Y|. As a consequence of this result, we obtain quasi-polynomial time algorithms for generating all maximal integer feasible solutions for a given monotone system of separable inequalities, for generating all p-inefficient points of a given discrete probability distribution, and for generating all maximal empty hyper-rectangles for a given set of points in ℝn. This provides a substantial improvement over previously known exponential algorithms for these generation problems related to Integer and Stochastic Programming, and Data Mining. Furthermore, we give an incremental polynomial time generation algorithm for monotone systems with fixed number of separable inequalities, which, for the very special case of one inequality, implies that for discrete probability distributions with independent coordinates, both p-efficient and p-inefficient points can be separately generated in incremental polynomial time.

UR - http://www.scopus.com/inward/record.url?scp=26844553656&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=26844553656&partnerID=8YFLogxK

M3 - Article

VL - 2719

SP - 543

EP - 555

JO - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

JF - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SN - 0302-9743

ER -