Abstract
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.
The research of the first and third authors was supported in part by the Office of Naval Research (Grant N00014-92-J-1375), and the National Science Foundation (Grant DMS 98-06389). The research of the third and forth authors was supported in part by the National Science Foundation (Grant CCR-9618796).
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Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L., Makino, K. (2001). On Generating All Minimal Integer Solutions for a Monotone System of Linear Inequalities. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds) Automata, Languages and Programming. ICALP 2001. Lecture Notes in Computer Science, vol 2076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48224-5_8
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DOI: https://doi.org/10.1007/3-540-48224-5_8
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