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On Fourier's algorithm for linear arithmetic constraints


In the 1820s Fourier provided the first algorithm for solving linear arithmetic constraints. In other words, this algorithm determines whether or not the polyhedral set associated with the constraints is empty. We show here that Fourier's algorithm has an important hidden property: in effect it also computes the affine hull of the polyhedral set. This result is established by making use of a recent theorem on the independence of negative constraints.

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  1. 1.

    Adler, I., ‘Equivalent linear programs’, Technical Report, Department of Operations Research, University of California at Berkeley (1976).

  2. 2.

    DantzigG. B. and EavesB. C., ‘Fourier-Motzkin elimination and its dual’, J. Combinational Theory Ser. A, 14, 288–297 (1973).

  3. 3.

    DuffinR. J., ‘On Fourier's analysis of linear inequality systems’, Math. Programming Study 1, 71–95 (1974).

  4. 4.

    FourierJ.-B. J., reported in: ‘Analyse des travaux de l'Academie Royale des Sciences, pendant l'annee 1824, Partie mathematique’, Histoire de l'Academie Royale des Sciences de l'Institut de France 7, xlvii-lv (1827) (Partial English translation in: D. A. Kohler, ‘Translation of a report by Fourier on his work on linear inequalities’, Opsearch 10, 38–42 (1973).

  5. 5.

    Irigoin, F. and Triolet, R., ‘Supernode partitioning’, Proc. 15th POPL, pp. 319–329 (1988).

  6. 6.

    JaffarJ., MichaylovS., StuckeyP. and YapR., ‘The CLP (ℜ) language and system’, ACM Transactions on Programming Languages and Systems 14(3), 339–395 (July 1992).

  7. 7.

    Lassez, J.-L., ‘Parametric queries, linear constraints and variable elimination’, Proc. Conference on Design and Implementation of Symbolic Computer Systems, Springer-Verlag Lecture Notes in Computer Science 429, pp. 164–173 (1990).

  8. 8.

    LassezJ.-L. and McAloonK., ‘A canonical form for generalized linear constraints’, J. Symbolic Computation 13, 1–24 (1992).

  9. 9.

    Lassez, J.-L., Huynh, T. and McAloon, K., ‘Simplification and elimination of redundant linear arithmetic constraints’, Proc. North American Conference on Logic Programming, pp. 37–51 (1989).

  10. 10.

    Schrijver, A., Theory of Linear and Integer Programming, Wiley (1986).

  11. 11.

    TelgenJ., ‘Minimal representation of convex polyhedral sets’, J. Optimization Theory and Application, 38, 1–24 (1982).

  12. 12.

    Williams, H. P., ‘Fourier's method of linear programming and its dual’, Amer. Math. Monthly, 681–695 (November 1986).

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Lassez, J., Maher, M.J. On Fourier's algorithm for linear arithmetic constraints. Journal of Automated Reasoning 9, 373–379 (1992).

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Key words

  • Linear arithmetic
  • Fourier's algorithm
  • implicit equality
  • affine hull