Avoiding slack variables in the solving of linear diophantine equations and inequations

Theoretical Computer Science

Volumn 173, Issue 1, 1997, Pages 183-208

  Ajili, Farid ^a   Contejean, Evelyne ^b  

^a INRIA   (France)

^b UFR 919 Laboratoire d'Informatique Pour la Mécanique et les Sciences de l'Ingénieur   (France)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; COMPUTER SCIENCE; CONSTRAINT THEORY; LOGIC PROGRAMMING; SET THEORY;

BOUNDED STACK; INEQUATIONS; LINEAR DIOPHANTINE EQUATIONS; SLACK VARIABLES;

PROBLEM SOLVING;

EID: 0031075065     PISSN: 03043975     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-3975(96)00195-8     Document Type: Article

References (31)

1. H. Abdulrab and M. Maksimenko, General solutions of systems of linear diophantine equations and inequations, in: J. Hsiang, ed., 6th Internat. Conf. on Rewriting Techniques and Applications, (Kaiserslautern, Germany, 1995) Lecture Notes in Computer Science, Vol. 914 (Springer, Berlin, 1995).
2. H. Abdulrab and M. Maksimenko, General solutions of systems of linear diophantine equations and inequations, in: J. Hsiang, ed., 6th Internat. Conf. on Rewriting Techniques and Applications, (Kaiserslautern, Germany, 1995) Lecture Notes in Computer Science, Vol. 914 (Springer, Berlin, 1995).
3. F. Ajili, Etude de la résolution de contraintes diophantiennes linéaires sur les entiers naturels, Rapport de dea, Université Henri Poincaré - Nancy 1, 1994.
4. F. Ajili and E. Contejean, Complete solving of linear and diophantine equational and inequational systems without adding variables, Tech. Report 0175, INRIA, 1995.
5. F. Ajili, E. Contejean, E. Domenjoud, M. Filgueiras, C. Kirchner and A.-P. Tomás, Solving linear diophantine equations: The state of the art, in preparation, 1995.
6. A. Boudet, Evelyne Contejean and Hervé Devie, A new AC-unification algorithm with a new algorithm for solving diophantine equations, in: Proc. 5th IEEE Symp. Logic in Computer Science, Philadelphia (IEEE Computer Soc. Press, Silver Spring, MD, 1990) 289-299.
7. T.J. Chou and G.E. Collins, Algorithms for the solution of systems of linear diophantine equations, SIAM J. comput. 11 (1982) 687-708.
8. M. Clausen and A. Fortenbacher, Efficient solution of linear diophantine equations, J. Symbolic Comput. 8 (1989) 201-216.
9. A.H. Clifford and G.B. Preston, The Algebraic Theory of Semigroups, Mathematical Surveys, No. 7 (American Mathematical Society, Providence, RI, 1961). There are two volumes. The second was published in 1967.
10. E. Contejean, Solving *-problems modulo distributivity by a reduction to AC1-unification, J. Symbolic Comput. 16 (1993) 493-521.
11. E. Contejean and H. Devie, An efficient algorithm for solving systems of diophantine equations, Inform. Comput. 113 (1) (1994) 143-172.
12. E. Domenjoud, Outils pour la déduction automatique dans les théories associatives-commutatives, Thèse de doctorat de l'université de Nancy I, 1991.
13. E. Domenjoud, Solving systems of linear diophantine equations: an algebraic approach, in: Proc. 16th Mathematical Foundations of Computer Science, Warsaw, Lecture Notes in Computer Science, Vol. 520 (Springer, Berlin, 1991).
14. E. Domenjoud and A.P. Tomás, From Elliot-Mac Mahon to an algorithm for general linear constraints on naturals, in: U. Montanari and F. Rossi, eds., Principles and Practice of Constraint Programming, Lecture Notes in Computer Science, Vol. 976 (Springer, Berlin, 1995).
15. M. Filgueiras and A.P. Tomás, Fast methods for solving linear diophantine equations, in: M. Filgueiras and L. Damas, eds. Proc. 6th Portuguese Conf. on Artificial Intelligence, Lecture Notes in Artificial Intelligence, Vol. 727 (Springer, Berlin, 1993) 297-306.
16. T. Guckenbiehl and A. Herold, Solving linear diophantine equations, Tech. Report SEKI-85-IV-KL, 1985.
17. A. Herold and J.H. Siekmann, Unification in abelian semi-groups, J. Automat. Reason. 3 (1987) 247-283.
18. G. Huet, An algorithm to generate the basis of solutions to homogeneous linear diophantine equations, Inform. Process. Lett. 7 (3) (1978).
19. N. Karmarkar, A new polynomial algorithm for linear programming, in: Proc. 16th Ann. ACM Symp. on Theory of Computing (ACM, New York, 1984) 302-311; revised version: Combinatorica 4 (1984) 373-395.
20. N. Karmarkar, A new polynomial algorithm for linear programming, in: Proc. 16th Ann. ACM Symp. on Theory of Computing (ACM, New York, 1984) 302-311; revised version: Combinatorica 4 (1984) 373-395.
21. L.G. Khachiyan, Polynomial algorithms in linear programming, Zh. Vychisl. Mat. i Matematicheskoi Fiziki 20 (1980) 51-68.
22. C. Kirchner, From unification in combination of equational theories to a new AC unification algorithm, in: H. Ait-Kaci and M. Nivat, eds., Proc. Coll. on Resolution of Equations in Algebraic Structures (Academic Press, New York, 1987) 171-210.
23. J.L. Lambert, Une borne pour les générateurs des solutions entières positives d'une équation diophantienne linéaire, Comptes Rendus de l'Académie des Sciences de Paris, Serie I, 305 (1987) 39, 40.
24. G.S. Makanin, Algorithmic decidability of the rank of constant free equations in a free semigroup, Dokl. Akad. Nauk. SSSR 243 (1978) 243.
25. L. Pottier, Minimal solutions of linear diophantine systems: bounds and algorithms, in: Proc. 4th Internat. Conf. on Rewriting Techniques and Applications, Como, Italy (1991) 162-173.
26. J.F. Romeuf, A polynomial algorithm for solving systems of two linear diophantine equations, Tech. Report, Laboratoire d'Informatique de Rouen et LITP, France, 1989.
27. A. Schrijver Theory of Linear and Integer Programming (Wiley, New York, 1986).
28. J.C. Sogno, Analysis of standard and new algorithms for the integer and linear constraint satisfaction problem, Tech. Report, INRIA, 1992.
29. M. Stickel, A unification algorithm for associative-commutative functions, J. ACM 28 (1981) 423-434.
30. P. Van Hentenryck, Constraint Satisfaction in Logic Programming (MIT Press, Cambridge, MA, 1989).
31. P. Van. Hentenryck, H. Simonis and M. Dincbas, Constraint satisfaction using constraint logic programming, Artificial Intelligence 58 (1-3) (1992) 113-159.