Bounds on generalized Frobenius numbers

Let $N \geq 2$ and let $1 < a_1 < ... < a_N$ be relatively prime integers. The Frobenius number of this $N$-tuple is defined to be the largest positive integer that has no representation as $\sum_{i=1}^N a_i x_i$ where $x_1,...,x_N$ are non-negative integers. More generally, the $s$-Frobenius number is defined to be the largest positive integer that has precisely $s$ distinct representations like this. We use techniques from the Geometry of Numbers to give upper and lower bounds on the $s$-Frobenius number for any nonnegative integer $s$.Comment: We include an appendix with an erratum and addendum to the published version of this paper: two inaccuracies in the statement of Theorem 2.2 are corrected and additional bounds on s-Frobenius numbers are derive

Integer Points in Knapsack Polytopes and s-covering Radius

Given an integer matrix A satisfying certain regularity assumptions, we consider for a positive integer s the set F_s(A) of all integer vectors b such that the associated knapsack polytope P(A,b)={x: Ax=b, x non-negative} contains at least s integer points. In this paper we investigate the structure of the set F_s(A) sing the concept of s-covering radius. In particular, in a special case we prove an optimal lower bound for the s-Frobenius number

Generalized Frobenius Number of Three Variables

For $k \geq 2$, we let $A = (a_{1}, a_{2}, \ldots, a_{k})$ be a $k$-tuple of positive integers with $\gcd(a_{1}, a_2, \ldots, a_k) =1$ and, for a non-negative integer $s$, the generalized Frobenius number of $A$, $g(A;s) = g(a_1, a_2, \ldots, a_k;s)$, the largest integer that has at most $s$ representations in terms of $a_1, a_2, \ldots, a_k$ with non-negative integer coefficients. In this article, we give a formula for the generalized Frobenius number of three positive integers $(a_1,a_2,a_3)$ with certain conditions.Comment: 13 pages, comments welcom

Parametric Polyhedra with at least kk Lattice Points: Their Semigroup Structure and the k-Frobenius Problem

Given an integral $d \times n$ matrix $A$, the well-studied affine semigroup \mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\} can be stratified by the number of lattice points inside the parametric polyhedra $P_A(b)=\{x: Ax=b, x\geq0\}$. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset \mbox{ Sg}_{\geq k}(A) of all vectors b \in \mbox{ Sg}(A) such that $P_A(b) \cap {\mathbb Z}^n$ has at least $k$ solutions. We demonstrate that this set is finitely generated, it is a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computations. Related results can be derived for those right-hand-side vectors $b$ for which $P_A(b) \cap {\mathbb Z}^n$ has exactly $k$ solutions or fewer than $k$ solutions. (2) A computational complexity theory. We show that, when $n$, $k$ are fixed natural numbers, one can compute in polynomial time an encoding of \mbox{ Sg}_{\geq k}(A) as a multivariate generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors of bounded norm that have at least $k$ solutions. (3) Applications and computation for the $k$-Frobenius numbers. Using Generating functions we prove that for fixed $n,k$ the $k$-Frobenius number can be computed in polynomial time. This generalizes a well-known result for $k=1$ by R. Kannan. Using some adaptation of dynamic programming we show some practical computations of $k$-Frobenius numbers and their relatives

The generalized Frobenius problem via restricted partition functions

Given relatively prime positive integers, $a_1,\ldots,a_n$, the Frobenius number is the largest integer with no representations of the form $a_1x_1+\cdots+a_nx_n$ with nonnegative integers $x_i$. This classical value has recently been generalized: given a nonnegative integer $k$, what is the largest integer with at most $k$ such representations? Other classical values can be generalized too: for example, how many nonnegative integers are representable in at most $k$ ways? For sufficiently large $k$, we give a complete answer to these questions by understanding how the output of the restricted partition function (the function $f(t)$ giving the number of representations of $t$) "interlaces" with itself. Furthermore, we give the full asymptotics of all of these values, as well as reprove formulas for some special cases (such as the $n=2$ case and a certain extremal family from the literature). Finally, we obtain the first two leading terms of the restricted partition function as a so-called quasi-polynomial.Comment: 18 page

On the distance between Frobenius numbers

Let n ≥ 2 and k ≥ 1 be integers and a = (a_1,...,a_n) be an integer vector with positive coprime entries. The k-Frobenius number F_k(a) is the largest integer that cannot be represented as a nonnegative integer combination of a_i in at least k different ways. We study the quantity (F_k(a) − F_1(a))(a1···an)^(−1/(n−1)) and use obtained results to improve existing upper bounds for 2-Frobenius numbers. The proofs are based on packing and covering results from the geometry of numbers

On the distance between Frobenius numbers

Let n ≥ 2 and k ≥ 1 be integers and a = (a_1,...,a_n) be an integer vector with positive coprime entries. The k-Frobenius number F_k(a) is the largest integer that cannot be represented as a nonnegative integer combination of a_i in at least k different ways. We study the quantity (F_k(a) − F_1(a))(a1···an)^(−1/(n−1)) and use obtained results to improve existing upper bounds for 2-Frobenius numbers. The proofs are based on packing and covering results from the geometry of numbers

Generalised Frobenius numbers: geometry of upper bounds, Frobenius graphs and exact formulas for arithmetic sequences

Given a positive integer vector {\ve a}=(a_{1},a_{2}\dots,a_k)^t with \bea 1< a_{1}<\cdots<a_{k}\, \quad \text{and}\quad \gcd(a_{1},\ldots,a_{k})=1 \,. \eea The Frobenius number of the vector {\ve a}, \frob_k({\ve a}), is the largest positive integer that cannot be represented as $\sum\limits_{i=1}^{k}a_{i}x_{i}$, where $x_{1},\ldots,x_{k}$ are nonnegative integers. We also consider a generalised Frobenius number, known in the literature as the $s$-Frobenius number, \frob_{s}(a_{1},a_{2},\ldots,a_{k}), which is defined to be the largest integer that cannot be represented as $\sum\limits_{i=1}^{k}a_{i}x_{i}$ in at least $s$ distinct ways. The classical Frobenius number corresponds to the case $s=1$. The main result of the thesis is the new upper bound for the $2$-Frobenius number, \be \label{equ:UB} \frob_2(a_{1},\ldots,a_{k})\leq \frob_1(a_{1},\ldots,a_{k}) +2\left(\frac {(k-1)!}{{2(k-1) \choose k-1}}\right)^{1/(k-1)} \left(a_{1}\cdots a_{k}\right)^{1/(k-1)}\,, \ee that arises from studying the bounds for the quantity \big(\frob_s({\ve a})-\frob_1({\ve a})\big)\left(a_{1}\cdots a_{k}\right)^{-1/(k-1)}\,. The bound (\ref{equ:UB}) is an improvement, for $s=2$, on a bound given by Aliev, Fukshansky and Henk \cite{aliev2011generalized}. Our proofs rely on the geometry of numbers. By using graph theoretic techniques, we also obtain an explicit formula for the $2$-Frobenius number of the arithmetic progression $a,a+d,\ldots a+nd$ (i.e. the $a_{i}$'s are in an arithmetic progression) with $\gcd(a,d)=1$ and $1\leq d<a$. \be \label{2} \frob_{2}(a,a+d,\ldots a+nd)=a\left\lfloor\frac{a}{n}\right\rfloor+d(a+1)\,, \quad n \in \{2,3\}. \ee % This result generalises Roperts's result \cite{Roberts} for the Frobenius number of general arithmetic sequences. In the course of our investigations we derive a formula for the shortest path and the distance between any two vertices of a graph associated with the positive integers $a_{1},\ldots,a_{k}$. Based on our results, we observe a new pattern for the $2$-Frobenius number of general arithmetic sequences $a,a+d,\dots,a+nd$, $\gcd(a,d)=1$, which we state as a conjecture. Part of this work has appeared in \cite{Alievdistance}