Generalized Frobenius numbers: bounds and average behavior

Aliev, Iskander; Fukshansky, Lenny; Henk, Martin

doi:10.4064/aa155-1-5

Cited by 10 publications

(16 citation statements)

References 21 publications

(31 reference statements)

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“…On the other hand, from the lower bound in Theorem 1.1 of [4] and the proof of Proposition 1 in [6] we conclude that for large enough…”

Section: Algorithm 1 Dp Algorithm For the K-frobenius Numbermentioning

confidence: 78%

“…We start by observing that there is an upper bound for the k-Frobenius number. Indeed Theorem 1.1 in [4] gives already an upper bound that is certainly smaller than M = k(n − 1)!a 1 a 2 · · · a n . The k-Frobenius number must be smaller and thus we will use M in the bounding box created in Theorem 5.…”

Section: Proof Of Corollarymentioning

confidence: 99%

“…For the case k > 1 the upper bound in Theorem 1.1 of [4], together with (21) implies that for any ε > 0 for large enough T A k (T ) ≤ 8 π + 2(k − 1) + ε .…”

Section: Algorithm 1 Dp Algorithm For the K-frobenius Numbermentioning

confidence: 99%

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Parametric polyhedra with at least k lattice points: Their semigroup structure and the k-Frobenius problem

Aliev

Loera

Louveaux

2016

Recent Trends in Combinatorics

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Given an integral d × n matrix A, the well-studied affine semigroup Sg(A) = {b : Ax = b, x ∈ Z n , x ≥ 0} can be stratified by the number of lattice points inside the parametric polyhedra P A (b) = {x : Ax = b, x ≥ 0}. 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 Sg ≥k (A) of all vectors b ∈ Sg(A) such that P A (b) ∩ 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-handside vectors b for which P A (b) ∩ 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 Sg ≥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.

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“…On the other hand, from the lower bound in Theorem 1.1 of [4] and the proof of Proposition 1 in [6] we conclude that for large enough…”

Section: Algorithm 1 Dp Algorithm For the K-frobenius Numbermentioning

confidence: 78%

Section: Proof Of Corollarymentioning

confidence: 99%

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Parametric polyhedra with at least k lattice points: Their semigroup structure and the k-Frobenius problem

Aliev

Loera

Louveaux

2016

Recent Trends in Combinatorics

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“…Let n ≥ 2 be an integer and let (1) 1 < a 1 < · · · < a n be relatively prime integers. We say that a positive integer t is representable by the n-tuple a := (a 1 , .…”

Section: Introductionmentioning

confidence: 99%

Positive semigroups and generalized Frobenius numbers over totally real number fields

Fukshansky

Shi

2020

Moscow J. Comb. Number Th.

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Frobenius problem and its many generalizations have been extensively studied in several areas of mathematics. We study semigroups of totally positive algebraic integers in totally real number fields, defining analogues of the Frobenius numbers in this context. We use a geometric framework recently introduced by Aliev, De Loera and Louveaux to produce upper bounds on these Frobenius numbers in terms of a certain height function. We discuss some properties of this function, relating it to absolute Weil height and obtaining a lower bound in the spirit of Lehmer's conjecture for algebraic vectors satisfying some special conditions. We also use a result of Borosh and Treybig to obtain bounds on the size of representations and number of elements of bounded height in such positive semigroups of totally real algebraic integers.

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“…More recently a k-feasibility generalization of the Frobenius number was introduced and studied by Beck and Robins [8]. They give formulas for n = 2 of the k-Frobenius number, but for general n and k only bounds on the k-Frobenius number F k (a) are available (see [3], [4] and [19]). …”

Section: Introductionmentioning

confidence: 99%

Integer Programs with Prescribed Number of Solutions and a Weighted Version of Doignon-Bell-Scarf’s Theorem

Aliev

Loera

Louveaux

2014

Integer Programming and Combinatorial Optimization

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Abstract. In this paper we study a generalization of the classical feasibility problem in integer linear programming, where an ILP needs to have a prescribed number of solutions to be considered solved. We first provide a generalization of the famous Doignon-Bell-Scarf theorem: Given an integer k, we prove that there exists a constant c(k, n), depending only on the dimension n and k, such that if a polyhedron {x : Ax ≤ b} contains exactly k integer solutions, then there exists a subset of the rows of cardinality no more than c(k, n), defining a polyhedron that contains exactly the same k integer solutions. The second contribution of the article presents a structure theory that characterizes precisely the set Sg ≥k (A) of all vectors b such that the problem Ax = b, x ≥ 0, x ∈ Z n , has at least k-solutions. We demonstrate that this set is finitely generated, a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computation. Similar results can be derived for those right-hand-side vectors that have exactly k solutions or fewer than k solutions. Finally we show that, when n, k are fixed natural numbers, one can compute in polynomial time an encoding of Sg ≥k (A) as a generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors that have exactly k solutions (similarly for at least k or less than k solutions). Under the same assumptions we prove that the k-Frobenius number can be computed in polynomial time.

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Generalized Frobenius numbers: bounds and average behavior

Cited by 10 publications

References 21 publications

Parametric polyhedra with at least k lattice points: Their semigroup structure and the k-Frobenius problem

Parametric polyhedra with at least k lattice points: Their semigroup structure and the k-Frobenius problem

Positive semigroups and generalized Frobenius numbers over totally real number fields

Integer Programs with Prescribed Number of Solutions and a Weighted Version of Doignon-Bell-Scarf’s Theorem

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