Abstract
This paper deals with the study of test sets of the knapsack problem and simultaneous diophantine approximation. The Graver test set of the knapsack problem can be derived from minimal integral solutions of linear diophantine equations. We present best possible inequalities that must be satisfied by all minimal integral solutions of a linear diophantine equation and prove that for the corresponding cone the integer analogue of Caratheodory's theorem applies when the numbers are divisible.
We show that the elements of the minimal Hilbert basis of the dual cone of all minimal integral solutions of a linear diophantine equation yield best approximations of a rational vector “from above”. A recursive algorithm for computing this Hilbert basis is discussed. We also outline an algorithm for determining a Hilbert basis of a family of cones associated with the knapsack problem. Keywords: knapsack problem, simultaneous diophantine approximation, diophantine equation, Hilbert basis, test sets.
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References
P. Conti, C. Traverso, Buchberger algorithm and integer programming, Proceedings AAECC-9 (New Orleans), Springer LNCS 539, 130–139 (1991).
W. Cook, J. Fonlupt, and A. Schrijver, An integer analogue of Caratheodory's theorem, J. Comb. Theory (B) 40, 1986, 63–70.
J.G. van der Corput, Über Systeme von linear-homogenen Gleichungen und Ungleichungen, Proceedings Koninklijke Akademie van Wetenschappen te Amsterdam 34, 368–371 (1931).
P. Diaconis, R. Graham, and B. Sturmfels, Primitive partition identities, Paul Erdös is 80, Vol. II, Janos Bolyai Society, Budapest, 1–20 (1995).
F.R. Giles and W.R. Pulleyblank, Total dual integrality and integer polyhedra, Lineare Algebra Appl. 25, 191–196 (1979).
P. Gordan, Über die Auflösung linearer Gleichungen mit reellen Coefficienten, Math. Ann. 6, 23–28 (1873).
J. E. Graver, On the foundations of linear and integer programming I, Mathematical Programming 8, 207–226 (1975).
J.L. Lambert, Une borne pour les générateurs des solutions entiéres postives d'une équation diophnatienne linéaire, C.R. Acad. Sci. Paris 305, Série I, 1987, 39–40.
H. E. Scarf, Neighborhood systems for production sets with indivisibilities, Econometrica 54, 507–532 (1986).
A. Sebö, Hilbert bases, Caratheodory's Theorem and combinatorial optimization, in Proc. of the IPCO conference, Waterloo, Canada, 1990, 431–455.
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© 1997 Springer-Verlag Berlin Heidelberg
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Henk, M., Weismantel, R. (1997). Test sets of the knapsack problem and simultaneous diophantine approximation. In: Burkard, R., Woeginger, G. (eds) Algorithms — ESA '97. ESA 1997. Lecture Notes in Computer Science, vol 1284. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63397-9_21
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DOI: https://doi.org/10.1007/3-540-63397-9_21
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