TY - JOUR

T1 - Generating all minimal integral solutions to AND-OR systems of monotone inequalities

T2 - Conjunctions are simpler than disjunctions

AU - Khachiyan, Leonid

AU - Boros, Endre

AU - Elbassioni, Khaled

AU - Gurvich, Vladimir

N1 - Funding Information:
The authors are grateful for the partial support by DIMACS, the National Science Foundation's Center for Discrete Mathematics and Theoretical Computer Science.

PY - 2008/6/6

Y1 - 2008/6/6

N2 - We consider monotone ∨, ∧-formulae φ of m atoms, each of which is a monotone inequality of the form fi (x) ≥ ti over the integers, where for i = 1, ..., m, fi : Zn {mapping} R is a given monotone function and ti is a given threshold. We show that if the ∨-degree of φ is bounded by a constant, then for linear, transversal and polymatroid monotone inequalities all minimal integer vectors satisfying φ can be generated in incremental quasi-polynomial time. In contrast, the enumeration problem for the disjunction of m inequalities is NP-hard when m is part of the input. We also discuss some applications of the above results in disjunctive programming, data mining, matroid and reliability theory.

AB - We consider monotone ∨, ∧-formulae φ of m atoms, each of which is a monotone inequality of the form fi (x) ≥ ti over the integers, where for i = 1, ..., m, fi : Zn {mapping} R is a given monotone function and ti is a given threshold. We show that if the ∨-degree of φ is bounded by a constant, then for linear, transversal and polymatroid monotone inequalities all minimal integer vectors satisfying φ can be generated in incremental quasi-polynomial time. In contrast, the enumeration problem for the disjunction of m inequalities is NP-hard when m is part of the input. We also discuss some applications of the above results in disjunctive programming, data mining, matroid and reliability theory.

KW - Dualization

KW - Generation algorithms

KW - Hypergraph transversals

KW - Linear inequalities

KW - Monotone systems of inequalities

KW - Polymatroid inequalities

KW - Transversal inequalities

UR - http://www.scopus.com/inward/record.url?scp=47549092148&partnerID=8YFLogxK

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U2 - 10.1016/j.dam.2007.04.018

DO - 10.1016/j.dam.2007.04.018

M3 - Article

AN - SCOPUS:47549092148

VL - 156

SP - 2020

EP - 2034

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 11

ER -