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

Aliev, Iskander; Henk, Martin; Linke, Eva

doi:10.37236/2887

Cited by 6 publications

(9 citation statements)

References 19 publications

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“…The k-Frobenius Problem, also called the generalized Frobenius problem, has been intensely studied in recent years, see e.g. [AHL13,FS11].…”

Section: Expansions Of Polytopesmentioning

confidence: 99%

On the Number of Integer Points in Translated and Expanded Polyhedra

Nguyen¹,

Pak²

2020

Discrete Comput Geom

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We prove that the problem of minimizing the number of integer points in parallel translations of a rational convex polytope in R 6 is NP-hard. We apply this result to show that given a rational convex polytope P ⊂ R 6 , finding the largest integer t s.t. the expansion tP contains fewer than k integer points is also NP-hard. We conclude that the Ehrhart quasi-polynomials of rational polytopes can have arbitrary fluctuations. Integer Point Minimization (IPM) Input:AParametric polytopes P b := {x ∈ R n : Ax ≤ b} were introduced by Kannan [Kan90], who gave a polynomial time algorithm for IPM with k = 0 and n bounded. For larger fixed values k, Aliev, De Loera and Louveaux [ADL16] proved that IPM is also polynomial time by employing the short generating functions technique by Barvinok and Woods [BW03] (see also [Bar06b,Bar08]). The following problem is an especially attractive special case:

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“…The k-Frobenius Problem, also called the generalized Frobenius problem, has been intensely studied in recent years, see e.g. [AHL13,FS11].…”

Section: Expansions Of Polytopesmentioning

confidence: 99%

On the Number of Integer Points in Translated and Expanded Polyhedra

Nguyen¹,

Pak²

2020

Discrete Comput Geom

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“…Since then, many number theorists have contributed to the topic (e.g., see [27,71] and the references there). Note also that Frobenius problem is highly related to what optimizers call the Knapsack problem and it has many applications (see [2,7,3,54] and references therein).…”

Section: Motivation Prior and Related Workmentioning

confidence: 99%

“…It is known that, in analogy with (10), Sg ≥k (A) can be decomposed into the set of all integer points in the interior of a certain translated cone and a complex complementary set. More recent results (see [7]) attempt to estimate the location of such a cone along the fixed direction v = A 1, where 1 is the all-1-vector, in the interior of cone(A). The choice of v as the direction vector is dated back to the paper of Khovanskii [55] for k = 1.…”

Section: Asymptotic Structure Of Sg ≥K (A)mentioning

confidence: 99%

“…Now one is interested on finding the largest number b that cannot be represented in more than k ways. Beck and Robins gave 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 [4], [7] and [48] for prior work). Since then, many number theorists have contributed to the topic (e.g., see [27,71] and the references there).…”

Section: Motivation Prior and Related Workmentioning

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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“…These ideas have also been extended to the more general s-Frobenius problem in [10] and [1]. A higher-dimensional analogue of the Frobenius problem has also been considered in the recent years by several authors, notably in [3], [4], and [5]. This note is inspired by the work of Aliev, De Loera and Louveaux [5].…”

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

Cited by 6 publications

References 19 publications

On the Number of Integer Points in Translated and Expanded Polyhedra

On the Number of Integer Points in Translated and Expanded Polyhedra

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

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