The multidimensional Frobenius problem

Amos, Jeffrey; Pascu, I; Ponomarenko, Vadim; Treviño, Enrique; Zhang, Yan

doi:10.2140/involve.2011.4.187

Cited by 4 publications

(6 citation statements)

References 14 publications

(10 reference statements)

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“…As mentioned in the introduction, it was also shown in [17] that the C +u−v is the unique maximal cone with the property that all points in the interior admit a non-negative integral representation, but infinitely many on the boundary do not have such a representation. This seems to be a very particular property of the case m = n − 1 as it was already pointed out in [4]. We want to conclude the paper with another example showing that, in general, we have more than one maximal cone.…”

Section: Proof Of Theorem 11supporting

confidence: 56%

See 1 more Smart Citation

Integer Points in Knapsack Polytopes and $s$-Covering Radius

Aliev¹,

Henk²,

Linke³

2013

Electron. J. Combin.

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Given a matrix A ∈ Z m×n satisfying certain regularity assumptions, we consider for a positive integer s the set Fs(A) ⊂ Z m of all vectors b ∈ Z m such that the associated knapsack polytope: Ax = b} contains at least s integer points. In this paper we investigate the structure of the set Fs(A) using the concept of s-covering radius. In particular, in the case m = 1 we prove an optimal lower bound for the s-Frobenius number, which is the largest integer b such that P (A, b) contains less than s integer points.2010 Mathematics Subject Classification. Primary: 90C10, 52C07, 11D07; Secondary: 90C27, 11H06.

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Section: Proof Of Theorem 11supporting

confidence: 56%

“…inclusion) translated cone h + C such that all integral points in its interior belong to F 1 (A), but there are (infinitely many) integral points on its boundary which are not in F 1 (A). In general, i.e., for 1 < m < n − 1, such a cone is not uniquely determined, see, for instance, [4] and Section 5.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Integer Points in Knapsack Polytopes and $s$-Covering Radius

Aliev¹,

Henk²,

Linke³

2013

Electron. J. Combin.

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“…, t k ∈ Q ≥0 such that r k+1 = t 1 r 1 + · · · + t k r k . From this it easily follows (2) implies (1). Assume that there exist q 1 , .…”

Section: Gluings and Conesmentioning

confidence: 86%

Frobenius vectors, Hilbert series and gluings of affine semigroups

Assi¹,

García-Sánchez²,

Ojeda³

2015

J. Commut. Algebra

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Let S1 and S2 be two affine semigroups and let S be the gluing of S1 and S2. Several invariants of S are then related to those of S1 and S2; we review some of the most important properties preserved under gluings. The aim of this paper is to prove that this is the case for the Frobenius vector and the Hilbert series. Applications to complete intersection affine semigroups are also given. On gluins of affine semigroupsIn this section we take a quick tour summarizing some of the more relevant results on the gluing of affine semigroups. We also introduce concepts and notations that will be used later on in the paper.An affine semigroup S is finitely generated submonoid of Z m for some positive integer m. If S ∩ (−S) = 0, that is to say S is reduced, it can be shown that it has a unique minimal system of generators (see for instance [24, Chapter 3]). The cardinality of the minimal generating system of S is known as the embedding dimension of S. Recall that each reduced affine semigroup can be embedded into N m for some m. In the following we will assume that our affine semigroups are submonoids of N m .Given an affine semigroup S ⊆ N m , denote by G(S) the group spanned by S, that is,Let A be the minimal generating system of S, and A = A 1 ∪ A 2 be a nontrivial partition of A. Let S i = A i (the monoid generated by A i ), i ∈ {1, 2}. Then S = S 1 + S 2 . We say that S is the gluing of S 1 and S 2 by d if • d ∈ S 1 ∩ S 2 and, • G(S 1 ) ∩ G(S 2 ) = dZ. We will denote this fact by S = S 1 + d S 2 .There are several properties that are preserved under gluings, and also some invariants of a gluing S 1 + d S 2 can be computed by knowing their values in S 1 and S 2 . We summarize some of them next.Assume that A = {a 1 , . . . , a k }. The monoid homomorphism ϕ : N k → S induced by e i → a i , i ∈ {1, . . . , k} is an epimorphism (where e i is the ith row of the k × k identity matrix). Thus S is isomorphic as a monoid to N k / ker ϕ, where ker ϕ is the kernel congruence of ϕ, that is, the set of pairs (a, b) ∈ N k ×N k with ϕ(a) = ϕ(b). A presentation of S is a system of generators of ker ϕ. A minimal presentation is a presentation such that none of its proper subsets is a presentation. All minimal presentations have the same (finite) cardinality (see for instance [24, Corollary 9.5]). Suppose that S = S 1 + d S 2 , with S i = A i , i ∈ {1, 2} and A = A 1 ∪ A 2 a nontrivial partition of A. We may assume without loss of generality that A 1 = {a 1 , . . . , a l } and A 2 = {a l+1 , . . . , a k }. According to [21, Theorem 1.4], if we know minimal presentations ρ 1 and ρ 2 of S 1 and S 2 , respectively, then ρ = ρ 1 ∪ ρ 2 ∪ {(a, b)}

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“…First notice that if we pick t greater than the maximal expression in the upper bound of (4), then (19) implies G Λa (t) > s. In addition, (28) implies that for s ≤ τ N −1 + 1, g s (a) satisfies (5). As for lower bounds on g s (a), if we pick…”

Section: Bounds On G S (A)mentioning

confidence: 99%

“…It should also be mentioned that other generalizations of the Frobenius number of different nature have also been considered by a variety of authors. In particular, see Chapter 6 of [1], as well as more recent works [4], [5], and [23], among others, for further information and references.…”

Section: Introductionmentioning

confidence: 99%

Bounds on generalized Frobenius numbers

Fukshansky

Schürmann²

2011

European Journal of Combinatorics

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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

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The multidimensional Frobenius problem

Cited by 4 publications

References 14 publications

Integer Points in Knapsack Polytopes and $s$-Covering Radius

Integer Points in Knapsack Polytopes and $s$-Covering Radius

Frobenius vectors, Hilbert series and gluings of affine semigroups

Bounds on generalized Frobenius numbers

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