Dual-bounded generating problems: Efficient and inefficient points for discrete probability distributions and sparse boxes for multidimensional data

Leonid Khachiyan, Endre Boros, Khaled Elbassioni, Vladimir Gurvich, Kazuhisa Makino

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

We show that {divides} X {divides} ≤ n {divides} Y {divides} must hold for two finite sets X, Y ⊂ Rn whenever they can be separated by a nonnegative linear function such that X is above Y and the componentwise minimum of any two distinct points in X is dominated by some point in Y. As a consequence, we obtain an incremental quasi-polynomial time algorithm 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 hyper-rectangles which contain a specified fraction of points of a given set in Rn. This provides a substantial improvement over previously known exponential time 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, implying that for discrete probability distributions with independent coordinates, both p-efficient and p-inefficient points can be separately generated in incremental polynomial time.

Original languageBritish English
Pages (from-to)361-376
Number of pages16
JournalTheoretical Computer Science
Volume379
Issue number3
DOIs
StatePublished - 15 Jun 2007

Keywords

  • Incremental generation
  • Maximal empty boxes
  • p-efficient points

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