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

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Automata, Languages and Programming (ICALP 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2076))

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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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Author information

Authors and Affiliations

  1. RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway, NJ, 08854-8003

    E. Boros & V. Gurvich

  2. Department of Computer Science, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8003

    K. Elbassioni & L. Khachiyan

  3. Division of Systems Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, 560-8531, Japan

    K. Makino

Authors
  1. E. Boros
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  2. K. Elbassioni
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  3. V. Gurvich
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  4. L. Khachiyan
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  5. K. Makino
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Editor information

Editors and Affiliations

  1. Departament de Llenguatges i Sistemes Informátics, Univ. Politècnica de Catalunya, C/Jordi Girona Salgado 1-3, 08034, Barcelona, Spain

    Fernando Orejas

  2. Computer Technology Institute (CTI), University of Patras, 61 Riga Feraiou street, 26221, Patras, Greece

    Paul G. Spirakis

  3. Institute of Information and Computing Sciences, Utrecht University, Padualaan 14, 3584, CH, Utrecht, The Netherlands

    Jan van Leeuwen

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© 2001 Springer-Verlag Berlin Heidelberg

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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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  • Print ISBN: 978-3-540-42287-7

  • Online ISBN: 978-3-540-48224-6

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