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2001
Springer

On Generating All Minimal Integer Solutions for a Monotone System of Linear Inequalities

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On Generating All Minimal Integer Solutions for a Monotone System of Linear Inequalities
We consider the problem of enumerating all minimal integer solutions of a monotone system of linear inequalities. We first show that for any monotone system of r linear inequalities in n variables, the number of maximal infeasible integer vectors is at most rn times the number of minimal integer solutions to the system. This bound is accurate up to a polylog(r) factor and leads to a polynomial-time reduction of the enumeration problem to a natural generalization of the well-known dualization problem for hypergraphs, in which dual pairs of hypergraphs are replaced by dual collections of integer vectors in a box. We provide a quasi-polynomial algorithm for the latter dualization problem. These results imply, in particular, that the problem of incrementally generating minimal integer solutions of a monotone system of linear inequalities can be done in quasi-polynomial time. Key words: Integer programming, complexity of incremental algorithms, dualization, quasi-polynomial time, monotone ...
Endre Boros, Khaled M. Elbassioni, Vladimir Gurvic
Added 29 Jul 2010
Updated 29 Jul 2010
Type Conference
Year 2001
Where ICALP
Authors Endre Boros, Khaled M. Elbassioni, Vladimir Gurvich, Leonid Khachiyan, Kazuhisa Makino
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