Skip to main content
Springer Nature Link
Log in

Max Horn SAT and the minimum cut problem in directed hypergraphs

  • Published:
Mathematical Programming Submit manuscript

Abstract

In this paper we consider the Maximum Horn Satisfiability problem, which is reduced to the problem of finding a minimum cardinality cut on a directed hypergraph. For the latter problem, we propose different IP formulations, related to three different definitions of hyperpath weight. We investigate the properties of their linear relaxations, showing that they define a hierarchy. The weakest relaxation is shown to be equivalent to the relaxation of a well known IP formulation of Max Horn SAT, and to a max-flow problem on hypergraphs. The tightest relaxation, which is a disjunctive programming problem, is shown to have integer optimum. The intermediate relaxation consists in a set covering problem with a possible exponential number of constraints. This latter relaxation provides an approximation of the convex hull of the integer solutions which, as proven by the experimental results given, is much tighter than the one known in the literature. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Gallo, G. Longo, S. Nguyen, S. Pallottino, Directed Hypergraphs and Applications, Discrete Applied Mathematics 42 (1993) 177–201.

    Google Scholar 

  2. D. Pretolani, Satisfiability and Hypergraphs, Ph.D. Thesis, TD-12/93, Dipartimento di Informatica, University of Pisa, Italy, 1993.

    Google Scholar 

  3. G. Gallo, G. Rago, A Hypergraphs Approach to Logical Inference for Datalog Formulae, TR 28/90, Dipartimento di Informatica, Università di Pisa.

  4. G. Rago, Optimization, Hypergraphs and Logical Inference, Ph.D. Thesis, TD-4/94, Dipartimento di Informatica, University of Pisa, Italy, 1993.

    Google Scholar 

  5. B. Jaumard, B. Simeone, On the complexity of the maximum satisfiability problem for Horn formulas, Information Processing Letters 26 (1987/1988) 1–4.

    Google Scholar 

  6. J. Kratochvíl, M. Křivánek, Satisfiability of co-nested formulas, Acta Informatica 30 (1993) 397–403.

    Google Scholar 

  7. M.X. Goemans, D.P. Williamson, New 3/4-approximation algorithms for the maximum satisfiability problem, SIAM Journal on Discrete Mathematics 7 (1994) 656–666.

    Google Scholar 

  8. M.X. Goemans, D.P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM 42 (1995) 1115–1145.

    Google Scholar 

  9. J.M. Bourjolly, P.L. Hammer, W.R. Pulleyblank, B. Simeone, Boolean Combinatorial Bounding of Maximum 2-Satisfiability, RRR 5-92, RUTCOR, Rutgers University, 1992.

  10. J. Cheriyan, W.H. Cunningham, L. Tuncel, Y. Wang, A Linear Programming and Rounding Approach to Max 2-SAT, in: D.S. Johnson, M.A. Trick (Eds.),Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 26, American Mathematical Society (1996).

  11. G. Gallo, R. Cambini, M.G. Scutellà, Flows on hypergraphs, Mathematical Programming 78 (1997) 195–217.

    Google Scholar 

  12. W.F. Dowling, J.H. Gallier, Linear-time algorithms for testing the satisfiability of propositional Horn formulae, Journal of Logic Programming 1 (1984) 267–284.

    Google Scholar 

  13. G. Ausiello, G.F. Italiano, U. Nanni, Dynamic maintenance of directed hypergraphs, Theoretical Computer Science 72 (1990) 97–117.

    Google Scholar 

  14. R.G. Jeroslow, K. Martin, R.L. Rardin, J. Wang, Gainfree Leontief substitution flow problems, Mathematical Programming 57 (3) (1992) 375–414.

    Google Scholar 

  15. R.G. Jeroslow, Logic-based decision support — mixed integer model formulation, Annals of Discrete Mathematics 40 (1989).

  16. R.G. Jeroslow, J. Wang, Dynamic programming, integral polyhedra and Horn clause knowledge base, ORSA Journal on Computing 1 (1989) 7–19.

    Google Scholar 

  17. G.L. Nemhauser, L.A. Wolsey, Integer and Combinatorial Optimization, Wiley, UK, 1988.

    Google Scholar 

  18. J.N. Hooker, V. Vinay, An empirical study of branching rules for satisfiability, presented at the third Symposium on Artificial Intelligence and Mathematics, January 1994.

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Pisa, Corso Italia 40, 156125, Pisa, Italy

    G. Gallo, C. Gentile, D. Pretolani & G. Rago

Authors
  1. G. Gallo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Gentile
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. Pretolani
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. G. Rago
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Corresponding author.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gallo, G., Gentile, C., Pretolani, D. et al. Max Horn SAT and the minimum cut problem in directed hypergraphs. Mathematical Programming 80, 213–237 (1998). https://doi.org/10.1007/BF01581727

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01581727

Keywords