Coin problem

About: Coin problem is a research topic. Over the lifetime, 202 publications have been published within this topic receiving 3455 citations.

The diophantine frobenius problem

Abstract: Preface Acknowledgements 1. Algorithmic Aspects 2. The Frobenius Number for Small n 3. The General Problem 4. Sylvester Denumerant 5. Integers without Representation 6. Generalizations and Related Problems 7. Numerical Semigroups 8. Applications of the Frobenius Number 9. Appendix A Bibliography

On the linear diophantine problem of Frobenius.

Abstract: The problem of Frobenius consists in determining the largest integer g(a1,a2,...,ak) with no such representation. To the author, a particularly nice aspect of this problem is the ease with which it can be explained to a non-mathematician : Given coins (in sufficient supply) of denominations ai9 a2, . . ., ak. Determine the largest amount which cannot be formed by means of these coins. It is well known that (1. 1) g(ai,a2) = ala2-al-a2. For k > 2, formulas for g have been proved only in special cases. More or less accurate bounds for g have also been given. One of these bounds, due to Erd s and Graham, is improved below. In later years, some authqrs have also examined the number

Lattice translates of a polytope and the Frobenius problem

Abstract: This paper considers the “Frobenius problem”: Givenn natural numbersa1,a2,...an such that their greatest common divisor is 1, find the largest natural number that is not expressible as a nonnegative integer combination of them. This problem can be seen to be NP-hard. For the casesn=2,3 polynomial time algorithms, are known to solve it. Here a polynomial time algorithm is given for every fixedn. This is done by first proving an exact relation between the Frobenius problem and a geometric concept called the “covering radius”. Then a polynomial time algorithm is developed for finding the covering radius of any polytope in a fixed number of dimensions. The last algorithm relies on a structural theorem proved here that describes for any polytopeK, the setK+ℤh={x∶x∈ℝn;x=y+z;y∈K;z∈ℤn} which is the portion of space covered by all lattice translates ofK. The proof of the structural theorem relies on some recent developments in the Geometry of Numbers. In particular, it uses a theorem of Kannan and Lovasz [11], bounding the width of lattice-point-free convex bodies and the techniques of Kannan, Lovasz and Scarf [12] to study the shapes of a polyhedron obtained by translating each facet parallel, to itself. The concepts involved are defined from first principles. In a companion paper [10], I extend the structural result and use that to solve a general problem of which the Frobenius problem is a special case.