Fast and Accurate Bounds on Linear Programs

  • Ernst Althaus
  • Daniel Dumitriu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5526)


We present an algorithm that certifies the feasibility of a linear program while using rational arithmetic as little as possible. Our approach relies on computing a feasible solution of the linear program that is as far as possible from satisfying an inequality at equality. To realize such an approach, we have to detect the set of inequalities that can only be satisfied at equality.

Compared to previous approaches for this problem our algorithm has a much higher rate of success.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Devulder, S., Lambert, J.L.: A comparative study between linear programming validation (LPV) and other verification methods. In: Automated Software Engineering (ASE), pp. 299–302 (1999)Google Scholar
  2. 2.
    Brinkmann, R., Drechsler, R.: RTL-Datapath verification using integer linear programming. In: Design Automation Conference (ASP-DAC), pp. 741–746. IEEE, Los Alamitos (2002)Google Scholar
  3. 3.
    Dellacherie, S., Devulder, S., Lambert, J.L.: Software verification based on linear programming. In: Woodcock, J.C.P., Davies, J., Wing, J.M. (eds.) FM 1999. LNCS, vol. 1709, pp. 1147–1165. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  4. 4.
    Dhiflaoui, M., Funke, S., Kwappik, C., Mehlhorn, K., Seel, M., Schömer, E., Schulte, R., Weber, D.: Certifying and repairing solutions to large LPs how good are LP-solvers? In: Symposium of Discrete Algorithms (SODA), pp. 255–256 (2003)Google Scholar
  5. 5.
    Koch, T.: The final NETLIB-LP results. Oper. Res. Lett. 32(2), 138–142 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Applegate, D., Cook, W., Dash, S., Espinoza, D.: Exact solutions to linear programming problems. Oper. Res. Lett. 35(6), 693–699 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Neumaier, A., Shcherbina, O.: Safe bounds in linear and mixed-integer linear programming. Math. Program. 99(2), 283–296 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Keil, C., Jansson, C.: Computational experience with rigorous error bounds for the netlib linear programming library. Reliable Computing 12(4), 303–321 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Netlib: A Linear Programming Library,
  10. 10.
    Higham, N.J.: A survey of condition number estimation for triangular matrices. SIAM Review 29(4), 575–596 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    SoPlex: The Sequential object-oriented simplex,
  12. 12.
    Wunderling, R.: Paralleler und objektorientierter Simplex-Algorithmus. Ph.D thesis, Technische Universität Berlin (1996),
  13. 13.
    GMP: The GNU Multiple Precision Arithmetic Library,
  14. 14.
    Boost: C++ Libraries,
  15. 15.
    Achterberg, T.: SCIP – a framework to integrate constraint and mixed integer programming. Technical Report 04-19, Zuse Institute Berlin (2004),
  16. 16.
    Fränzle, M., Herde, C., Ratschan, S., Schubert, T., Teige, T.: Efficient solving of large non-linear arithmetic constraint systems with complex boolean structure. JSAT Special Issue on Constraint Programming and SAT 1, 209–236 (2007)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Ernst Althaus
    • 1
  • Daniel Dumitriu
    • 1
  1. 1.Institut für InformatikJohannes Gutenberg UniversityMainzGermany

Personalised recommendations