Frobenius problem and the covering radius of a lattice

Let $N \geq2$ and let $1 < a_1 < ... < a_N$ be relatively prime integers. Frobenius number of this $N$-tuple is defined to be the largest positive integer that cannot be expressed as $\sum_{i=1}^N a_i x_i$ where $x_1,...,x_N$ are non-negative integers. The condition that $gcd(a_1,...,a_N)=1$ implies that such number exists. The general problem of determining the Frobenius number given $N$ and $a_1,...,a_N$ is NP-hard, but there has been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating Frobenius number to the covering radius of the null-lattice of this $N$-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.Comment: 12 pages; minor revisions; to appear in Discrete and Computational Geometr

An Optimal Lower Bound for the Frobenius Problem

Given $N$ positive integers $a_1, ..., a_N$ with $\gcd(a_1, ..., a_N)=1$, let $f_N$ denote the largest natural number which is not a positive integer combination of $a_1, ..., a_N$. This paper gives an optimal lower bound for $f_N$ in terms of the absolute inhomogeneous minimum of the standard $(N-1)$-simplex.Comment: 10 page

Revisiting the hexagonal lattice: on optimal lattice circle packing

In this note we give a simple proof of the classical fact that the hexagonal lattice gives the highest density circle packing among all lattices in $R^2$. With the benefit of hindsight, we show that the problem can be restricted to the important class of well-rounded lattices, on which the density function takes a particularly simple form. Our proof emphasizes the role of well-rounded lattices for discrete optimization problems.Comment: 8 pages, 1 figure; to appear in Elemente der Mathemati

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 Knapsacks: Average Behavior of the Frobenius Numbers

The main result of the paper shows that the asymptotic growth of the Frobenius number in average is significantly slower than the growth of the maximum Frobenius number

Well-rounded zeta-function of planar arithmetic lattices

We investigate the properties of the zeta-function of well-rounded sublattices of a fixed arithmetic lattice in the plane. In particular, we show that this function has abscissa of convergence at $s=1$ with a real pole of order 2, improving upon a recent result of S. Kuehnlein. We use this result to show that the number of well-rounded sublattices of a planar arithmetic lattice of index less or equal $N$ is $O(N \log N)$ as $N \to \infty$. To obtain these results, we produce a description of integral well-rounded sublattices of a fixed planar integral well-rounded lattice and investigate convergence properties of a zeta-function of similarity classes of such lattices, building on some previous results of the author.Comment: 12 pages; to appear in PAM

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

On well-rounded sublattices of the hexagonal lattice

We produce an explicit parameterization of well-rounded sublattices of the hexagonal lattice in the plane, splitting them into similarity classes. We use this parameterization to study the number, the greatest minimal norm, and the highest signal-to-noise ratio of well-rounded sublattices of the hexagonal lattice of a fixed index. This investigation parallels earlier work by Bernstein, Sloane, and Wright where similar questions were addressed on the space of all sublattices of the hexagonal lattice. Our restriction is motivated by the importance of well-rounded lattices for discrete optimization problems. Finally, we also discuss the existence of a natural combinatorial structure on the set of similarity classes of well-rounded sublattices of the hexagonal lattice, induced by the action of a certain matrix monoid.Comment: 21 pages (minor correction to the proof of Lemma 2.1); to appear in Discrete Mathematic

LLL-reduction for Integer Knapsacks

Given an integer mxn matrix A satisfying certain regularity assumptions, a well-known integer programming problem asks to find an integer point in the associated knapsack polytope P(A, b)={x: A x= b, x>=0} or determine that no such point exists. We obtain a LLL-based polynomial time algorithm that solves the problem subject to a constraint on the location of the vector b.Comment: improved versio