Difference between revisions of "Frobenius problem"
From Encyclopedia of Mathematics
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| − | Given | + | 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 | + | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110200/f1102009.png" /> there is a polynomial-time algorithm to solve the Frobenius problem, [[#References|[a1]]]. | For a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110200/f1102009.png" /> there is a polynomial-time algorithm to solve the Frobenius problem, [[#References|[a1]]]. | ||
Revision as of 15:25, 10 August 2014
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 
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 |
How to Cite This Entry:
Frobenius problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frobenius_problem&oldid=32825
Frobenius problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frobenius_problem&oldid=32825
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article