Skip to main content
SpringerLink
Log in

On a decision procedure for quantified linear programs

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

Quantified linear programming is the problem of checking whether a polyhedron specified by a linear system of inequalities is non-empty, with respect to a specified quantifier string. Quantified linear programming subsumes traditional linear programming, since in traditional linear programming, all the program variables are existentially quantified (implicitly), whereas, in quantified linear programming, a program variable may be existentially quantified or universally quantified over a continuous range. In this paper, the term linear programming is used to describe the problem of checking whether a system of linear inequalities has a feasible solution. On account of the alternation of quantifiers in the specification of a quantified linear program (QLP), this problem is non-trivial. QLPs represent a class of declarative constraint logic programs (CLPs) that are extremely rich in their expressive power. The complexity of quantified linear programming for arbitrary constraint matrices is unknown. In this paper, we show that polynomial time decision procedures exist for the case in which the constraint matrix satisfies certain structural properties. We also provide a taxonomy of quantified linear programs, based on the structure of the quantifier string and discuss the computational complexities of the constituent classes.

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

Access this article

Log in via an institution

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. Alur, R., Dill, D.L.: A theory of timed automata. Theor. Comput. Sci. 126(2), 183–235 (1994). Fundamental Study. 25 April

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Ben-Ayed, O.: Bilevel linear programming. Comput. Oper. Res. 20, 485–501 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  4. Benhamou, F., Goualard, F.: Universally quantified interval constraints. In: CP, pp. 67–82 (2000)

  5. Benhamou, F., Goulard, F.: Universally quantified interval constraints. In: Proceedings of Constraint Programming (2000) September

  6. Benhamou, F., Older, W.J.: Applying interval arithmetic to real, integer, and boolean constraints. J. Log. Program. 32(1), 1–24 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cadoli, M., Giovanardi, M., Giovanardi, A., Schaerf, M.: Experimental analysis of the computational cost of evaluating quantified boolean formulae. In: Lecture Notes in Artificial Intelligence (1997)

  8. Cadoli, M., Giovanardi, A., Schaerf, M.: An algorithm to evaluate quantified boolean formulae. In: AAAI-98 (1998) July

  9. 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 

  10. Chandru, V., Rao, M.R.: Linear programming. In: Algorithms and Theory of Computation Handbook. CRC Press, Boca Raton, FL (1999)

    Google Scholar 

  11. Choi, S.: Dynamic time-based scheduling for hard real-time systems. Ph.D. thesis, University of Maryland, College Park (1997) June

  12. Choi, S., Agrawala, A.K.: Dynamic dispatching of cyclic real-time tasks with relative timing constraints. Real-Time Syst. 19(1), 5–40 (2000)

    Article  Google Scholar 

  13. Colmerauer, A.: Logic programming and its applications, Chapter theoretical model of prologII. Ablex Series in Artificial Intelligence, pp. 181–200. Ablex Publishing Corporation (1986)

  14. Colmerauer, A., Dao, T.: Expressiveness of full first order constraints in the algebra of finite or infinite trees. In: Proceedings of Constraint Programming (2000) September

  15. Corbett, J.C., Avrunin, G.S.: Using integer programming to verify safety and liveness properties. Form. Methods Syst. Des. 6(1), 97–123 (1995)

    Article  MATH  Google Scholar 

  16. Dantzig, G.B., Eaves, B.C.: Fourier–Motzkin elimination and its dual. J. Comb. Theory (A) 14, 288–297 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  17. Donald, B., Kapur, D., Mundy, J.L.: Symbolic and numerical computation for artificial intelligence. Academic Press (1992)

  18. Fourier, J.B.J.: Reported in: Analyse de Travaux de l’Academie Royale des Sciences, Pendant l’annee 1824, Partie Mathematique, Histoire de ’Academie Royale de Sciences de l’Institue de France 7 (1827) xlvii-lv. (Partial English translation in: Kohler, D.A.: Translation of a report by Fourier on his work on linear inequalities. Opsearch 10, 38–42 (1973).). Academic Press (1824)

    Google Scholar 

  19. Gerber, R., Pugh, W., Saksena, M.: Parametric dispatching of hard real-time tasks. IEEE Trans. Comput. 44(3), 471–479 (1995)

    Article  MATH  Google Scholar 

  20. Hemaspaandra, L.A., Ogihara, M.: The complexity theory companion. Springer-Verlag, New York (2002)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. Hooker, J.N.: Logic-Based Methods for Optimization. Series in Discrete Mathematics and Optimization. John Wiley & Sons Inc. (2000)

  23. Huynh, T., Lassez, C., Lassez, J.-L.: Fourier algorithm revisited. In: Kirchner, H., Wechler, W. (eds.) Proceedings Second International Conference on Algebraic and Logic Programming. Lecture Notes in Computer Science, vol. 463, pp. 117–131. Nancy, France, Springer-Verlag (1990) October

  24. Lassez, C., Lassez, J.L.: Quantifier elimination for conjunctions of linear constraints via a convex hull algorithm. Academic Press (1993)

  25. Lassez, J.-L., Maher, M.: On Fourier’s algorithm for linear constraints. J. Autom. Reason. 9, 373–379 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  26. Levi, S.T., Tripathi, S.K., Carson, S.D., Agrawala, A.K.: The Maruti hard real-time operating system. ACM Spec. Interest Group on Oper. Syst. 23(3), 90–106 (1989) July

    Google Scholar 

  27. Loos, R., Weispfenning, V.: Applying linear quantifier elimination. Comput. J. 36(5), 450–462 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  28. Mosse, D., Ko, K.-T., Agrawala, A.K., Tripathi, S.K.: Maruti: an environment for hard real-time applications. In: Agrawala, A.K., Gordon, K.D., Hwang, P. (eds.) Maruti OS, pp. 75–85. IOS Press (1992)

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

    MATH  Google Scholar 

  30. Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization. Prentice Hall (1982)

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

    MATH  Google Scholar 

  32. Ratschan, S.: Applications of quantified constraint solving over the reals. http://www.cs.cas.cz/~ratschan/preprints.html

  33. Ratschan, S.: Continuous first-order constraint satisfaction. In: AISC, pp. 181–195 (2002)

  34. Ratschan, S., She, Z.: Safety verification of hybrid systems by constraint propagation based abstraction refinement. In: HSCC, pp. 573–589 (2005)

  35. Saksena, M.: Parametric scheduling in hard real-time systems. Ph.D. thesis, University of Maryland, College Park (1994) June

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

    Google Scholar 

  37. Subramani, K.: Duality in the parametric polytope and its applications to a scheduling problem. Ph.D. thesis, University of Maryland, College Park (2000) August

  38. Subramani, K.: Parametric scheduling – algorithms & complexity. In: Monien, B. et. al. (eds.) Proceedings of the 8th International Conference on High-Performance Computing (Hi-PC), vol. 2228 of Lecture Notes in Computer Science, pp. 36–46. Springer-Verlag (2001) December

  39. Subramani, K.: A specification framework for real-time scheduling. In: Grosky, W.I., Plasil, F. (eds.) Proceedings of the 29th Annual Conference on Current Trends in Theory and Practice of Informatics (SOFSEM). Lecture Notes in Computer Science, vol. 2540, pp. 195–207. Springer-Verlag (2002) November

  40. Tarski, A.: A Decision Method for Elementary Algebra and Geometry. Univ. of California Press, Berkeley (1951)

    MATH  Google Scholar 

  41. Vaidya, P.M.: An algorithm for linear programming which requires O(((m+n)n 2+(m+n)1.5 n)L) arithmetic operations. In: Aho, A. (ed.) Proceedings of the 19th Annual ACM Symposium on Theory of Computing, pp. 29–38. ACM Press, New York City, NY (1987) May

    Google Scholar 

  42. Vavasis, S.A.: Nonlinear Optimization: Complexity Issues. Oxford University Press, New York (1991)

    MATH  Google Scholar 

  43. Vicente, L., Calamai, P.: Bilevel and multilevel programming: a bibliography review. J. Global Optim. 5, 291–306 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  44. Weispfenning, V.: Quantifier elimination for real algebra – the quadratic case and beyond. Appl. Algebra Eng. Commun. Comput. 8(2), 85–101 (1997)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LDCSEE, West Virginia University, Morgantown, WV, USA

    K. Subramani

Authors
  1. K. Subramani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. Subramani.

Additional information

This research has been supported in part by the Air Force Office of Scientific Research under contract FA9550-06-1-0050.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Subramani, K. On a decision procedure for quantified linear programs. Ann Math Artif Intell 51, 55–77 (2007). https://doi.org/10.1007/s10472-007-9085-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10472-007-9085-y

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation