Frobenius problem

2020 Mathematics Subject Classification: Primary: 11D07 [MSN][ZBL]

coin-change problem

Given $n$ natural numbers $a_1,\ldots,a_n$ with greatest common divisor $1$, find the largest natural number that is not expressible as a linear non-negative integer combination of the $a_1,\ldots,a_n$.

For $n=2$ the answer is given by $a_1a_2-a_1-a_2$. The general problem is $\mathcal{NP}$-hard.

For a fixed $n$ there is a polynomial-time algorithm to solve the Frobenius problem, [a1].

The Frobenius problem is related to the study of maximal lattice-point-free convex bodies (in the geometry of numbers), [a2].

References

[a1]	R. Kannan, "Lattice translates of a polytope and the Frobenius problem" Combinatorica , 12 (1992) pp. 161–172
[a2]	L. Lovász, "Geometry of numbers and integer programming" M. Iri (ed.) K. Tanabe (ed.) , Mathematical Programming , Kluwer Acad. Publ. (1989) pp. 177–202