A Diophantine problem of Frobenius in terms of the least common multiple

Raczunas, Marek; Chrząstowski-Wachtel, Piotr

doi:10.1016/0012-365x(95)00199-7

Cited by 13 publications

(20 citation statements)

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“…In theorem 3.1 we give formulas for g(A), N ′ (A) and N (A) in terms of F p (t). As a consequence we obtain an upper bound for the Frobenius number (corollary 3.2) which improves the upper bound given by Chrzastowski-Wachtel and mentioned in [9]. A characterization of numerical symmetric and pseudo-symmetric semigroups (corollary 3.4) is also obtained.…”

Section: Introductionsupporting

confidence: 71%

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Frobenius Number of a Linear Diophantine Equation

Badra

2017

Commutative Ring Theory and Applications

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We denote by N0 the set of nonnegative integers. Let d ≥ 1 and A = {a1, . . . , a d } a set of positive integers. For every n ∈ N0, we write s(n) for the number of solutionsthe Frobenius number of A. Let S(A) be the subsemigroup of (N0, +) generated by A. We set S ′ (A) = N0\S(A), N ′ (A) = CardS ′ (A) and N (A) = Card S(A)∩{0, 1, .., g(A)}. Let p be a multiple of lcm(A) andWe give an upper bound for g(A) and reduction formulas for g(A), N ′ (A) and N (A). Characterizations of these invariants as well as numerical symmetric and pseudo-symmetric semigroups in terms of Fp(t), are also obtained.

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Section: Introductionsupporting

confidence: 71%

“…In theorem 3.7 we prove reduction formulas for g(A), N ′ (A) and N (A). The first one generalizes a Raczunas and Chrzastowski-Wachtel theorem [9]. As a consequence (corollary 3.10) we obtain a generalization of a Rödseth formula [10].…”

Section: Introductionmentioning

confidence: 67%

Frobenius Number of a Linear Diophantine Equation

Badra

2017

Commutative Ring Theory and Applications

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“…In practice, however, these bounds are generally far less accurate than U 2,3 (A). Bounds based on least common multiples [RC96,BB01] are even less accurate.…”

Section: Comparisons To Other Upper Boundsmentioning

confidence: 99%

Faster Algorithms for Frobenius Numbers

Beihoffer¹,

Hendry

Nijenhuis

et al. 2005

Electron. J. Combin.

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The Frobenius problem, also known as the postage-stamp problem or the money-changing problem, is an integer programming problem that seeks nonnegative integer solutions to $ x_1 a_1 +\cdots + x_n a_n =M$, where $ a_i $ and $M$ are positive integers. In particular, the Frobenius number $f(A)$, where $A=\{ a_i \}$, is the largest $M$ so that this equation fails to have a solution. A simple way to compute this number is to transform the problem to a shortest-path problem in a directed weighted graph; then Dijkstra's algorithm can be used. We show how one can use the additional symmetry properties of the graph in question to design algorithms that are very fast. For example, on a standard desktop computer, our methods can handle cases where $n=10$ and $ a_1 = 10^7$. We have two main new methods, one based on breadth-first search and another that uses the number theory and combinatorial structure inherent in the problem to speed up the Dijkstra approach. For both methods we conjecture that the average-case complexity is $O( a_1 \sqrt{n} )$. The previous best method is due to Böcker and Lipták and runs in time $O( a_1 n)$. These algorithms can also be used to solve auxiliary problems such as (1) find a solution to the main equation for a given value of $M$; or (2) eliminate all redundant entries from a basis. We then show how the graph theory model leads to a new upper bound on $f(A)$ that is significantly lower than existing upper bounds. We also present a conjecture, supported by many computations, that the expected value of $f(A)$ is a small constant multiple of $ \left({1\over2} \, n! \, \Pi A\right)^{1/(n-1)} -\Sigma A$.

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“…Our starting point in the construction of the bridge mentioned above is to show that S M appears naturally in the classical theory of numerical semigroups. Indeed, Raczunas and Chrzastowski-Wachtel in [RChW96] characterized certain semigroups, which realizes a sharp upper bound estimate for the Diophantine Frobenius problem/number. They called them 'strongly flat semigroups'.…”

Section: Introductionmentioning

confidence: 86%

On the geometry of strongly flat semigroups and their generalizations

Tamás¹,

Némethi²

2020

A Panorama of Singularities

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Our goal is to convince the readers that the theory of complex normal surface singularities can be a powerful tool in the study of numerical semigroups, and, in the same time, a very rich source of interesting affine and numerical semigroups. More precisely, we prove that the strongly flat semigroups, which satisfy the maximality property with respect to the Diophantine Frobenius problem, are exactly the numerical semigroups associated with negative definite Seifert homology spheres via the possible 'weights' of the generic S 1 -orbit. Furthermore, we consider their generalization to the Seifert rational homology sphere case and prove an explicit (up to a Laufer computation sequence) formula for their Frobenius number. The singularities behind are the weighted homogeneous ones, whose several topological and analytical properties are exploited.

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A Diophantine problem of Frobenius in terms of the least common multiple

Cited by 13 publications

References 5 publications

Frobenius Number of a Linear Diophantine Equation

Frobenius Number of a Linear Diophantine Equation

Faster Algorithms for Frobenius Numbers

On the geometry of strongly flat semigroups and their generalizations

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