TY - CHAP

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

Y1 - 2003

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.

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U2 - 10.1007/3-540-45061-0_44

DO - 10.1007/3-540-45061-0_44

M3 - Chapter

AN - SCOPUS:26844553656

SN - 3540404937

SN - 9783540404934

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 543

EP - 555

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

A2 - Baeten, Jos C. M.

A2 - Lenstra, Jan Karel

A2 - Parrow, Joachim

A2 - Woeginger, Gerhard J.

PB - Springer Verlag

ER -