Skip to main content
Springer Nature Link
Log in

Analyzing Selected Quantified Integer Programs

  • Conference paper
Automated Reasoning (IJCAR 2004)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 3097))

Included in the following conference series:

Abstract

In this paper, we introduce a problem called Quantified Integer Programming, which generalizes the Quantified Satisfiability problem (QSAT). In a Quantified Integer Program (QIP), the program variables can assume arbitrary integral values, as opposed to the boolean values that are assumed by the variables of an instance of QSAT. QIPs naturally represent 2-person integer matrix games. The Quantified Integer Programming problem is PSPACE-hard in general, since the QSAT problem is PSPACE-complete. We focus on analyzing various special cases of the general problem, with a view to discovering subclasses that are tractable. Subclasses of the general QIP problem are obtained by restricting either the constraint matrix or the quantifier specification. We show that if the constraint matrix is totally unimodular, the problem of deciding a QIP can be solved in polynomial time.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms and Applications. Prentice-Hall, Englewood Cliffs (1993)

    Google Scholar 

  2. Aspvall, B., Plass, M.F., Tarjan, R.: A linear-time algorithm for testing the truth of certain quantified boolean formulas. Information Processing Letters 8(3), 121–123 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cadoli, M., Giovanardi, A., Schaerf, M.: An algorithm to evaluate quantified boolean formulae. In: AAAI 1998 (July 1998)

    Google Scholar 

  4. Chandru, V., Hooker, J.N.: Optimization Methods for Logical Inference. Series in Discrete Mathematics and Optimization. John Wiley & Sons Inc., New York (1999)

    Google Scholar 

  5. Gavril, F.: An efficiently solvable graph partition, problem to which many problems are reducible. Information Processing Letters 45(6), 285–290 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  6. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman Company, San Francisco (1979)

    MATH  Google Scholar 

  7. Hochbaum, D.S., Naor, J.(Seffi).: Simple and fast algorithms for linear and integer programs with two variables per inequality. SIAM Journal on Computing 23(6), 1179–1192 (1994)

    Google Scholar 

  8. Hirschberg, D.S., Wong, C.K.: A polynomial algorithm for the knapsack problem in 2 variables. JACM 23(1), 147–154 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  9. Iwata, S., Matsui, T., McCormick, S.T.: A fast bipartite network flow algorithm for selective assembly. OR Letters, 137–143 (1998)

    Google Scholar 

  10. Johnson, D.S.: Personal Communication

    Google Scholar 

  11. Kannan, R.: A polynomial algorithm for the two-variable integer programming problem. JACM 27(1), 118–122 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lagarias, J.C.: The computational complexity of simultaneous Diophantine approximation problems. SIAM Journal on Computing 14(1), 196–209 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  13. McCormick, S.T.: Fast algorithms for parametric scheduling come from extensions to parametric maximum flow. Operations Research 47, 744–756 (2000)

    Article  MathSciNet  Google Scholar 

  14. Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. John Wiley & Sons, New York (1999)

    MATH  Google Scholar 

  15. Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, New York (1994)

    MATH  Google Scholar 

  16. Pinedo, M.: Scheduling: theory, algorithms, and systems. Prentice-Hall, Englewood Cliffs (1995)

    MATH  Google Scholar 

  17. Pugh, W.: The definition of dependence distance. Technical Report CSTR-2292, Dept. of Computer Science, Univ. of Maryland, College Park (November 1992)

    Google Scholar 

  18. Pugh, W.: The omega test: A fast and practical integer programming algorithm for dependence analysis. Comm. of the ACM 35(8), 102–114 (1992)

    Article  Google Scholar 

  19. Revesz, P.: Safe query languages for constraint databases. ACM Transactions on Database Systems 23(1), 58–99 (1998)

    Article  Google Scholar 

  20. Revesz, P.: The evaluation and the computational complexity of datalog queries of boolean constraints. International Journal of Algebra and Computation 8(5), 553–574 (1998)

    Article  MathSciNet  Google Scholar 

  21. Revesz, P.: Safe datalog queries with linear constraints. In: Maher, M.J., Puget, J.-F. (eds.) CP 1998. LNCS, vol. 1520, pp. 355–369. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  22. Schaefer, T.J.: The complexity of satisfiability problems. In: Aho, A. (ed.) Proceedings of the 10th Annual ACM Symposium on Theory of Computing, pp. 216–226. ACM Press, New York City (1978)

    Chapter  Google Scholar 

  23. Schrijver, A.: Theory of Linear and Integer Programming. John Wiley and Sons, New York (1987)

    Google Scholar 

  24. Stankovic, J.A., Spuri, M., Ramamritham, K., Buttazzo, G.C. (eds.): Deadline Scheduling for Real-Time Systems. Kluwer Academic Publishers, Dordrecht (1998)

    MATH  Google Scholar 

  25. Subramani, K.: On identifying simple and quantified lattice points in the 2SAT polytope. In: Calmet, J., Benhamou, B., Caprotti, O., Hénocque, L., Sorge, V., et al. (eds.) AISC 2002 and Calculemus 2002. LNCS (LNAI), vol. 2385, p. 217. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  26. Subramani, K.: A specification framework for real-time scheduling. In: Grosky, W.I., Plášil, F. (eds.) SOFSEM 2002. LNCS, vol. 2540, pp. 195–207. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  27. Subramani, K.: An analysis of quantified linear programs. In: Calude, C.S., Dinneen, M.J., Vajnovszki, V. (eds.) DMTCS 2003. LNCS, vol. 2731, pp. 265–277. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  28. van Hentenryck, P.: Personal Communication

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LCSEE, West Virginia University, Morgantown, WV

    K. Subramani

Authors
  1. K. Subramani
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Information Security Group, Dep. of Computer Science, ETH, Zurich, Switzerland

    David Basin

  2. LORIA & INRIA Lorraine UMR 7503,  

    Michaël Rusinowitch

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Subramani, K. (2004). Analyzing Selected Quantified Integer Programs. In: Basin, D., Rusinowitch, M. (eds) Automated Reasoning. IJCAR 2004. Lecture Notes in Computer Science(), vol 3097. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-25984-8_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-25984-8_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22345-0

  • Online ISBN: 978-3-540-25984-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics