71 research outputs found

    An Optimal Lower Bound for the Frobenius Problem

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    Given NN positive integers a1,...,aNa_1, ..., a_N with gcd⁑(a1,...,aN)=1\gcd(a_1, ..., a_N)=1, let fNf_N denote the largest natural number which is not a positive integer combination of a1,...,aNa_1, ..., a_N. This paper gives an optimal lower bound for fNf_N in terms of the absolute inhomogeneous minimum of the standard (Nβˆ’1)(N-1)-simplex.Comment: 10 page

    Normal Toric Ideals of Low Codimension

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    Every normal toric ideal of codimension two is minimally generated by a Grobner basis with squarefree initial monomials. A polynomial time algorithm is presented for checking whether a toric ideal of fixed codimension is normal

    High-Multiplicity Fair Allocation Using Parametric Integer Linear Programming

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    Using insights from parametric integer linear programming, we significantly improve on our previous work [Proc. ACM EC 2019] on high-multiplicity fair allocation. Therein, answering an open question from previous work, we proved that the problem of finding envy-free Pareto-efficient allocations of indivisible items is fixed-parameter tractable with respect to the combined parameter "number of agents" plus "number of item types." Our central improvement, compared to this result, is to break the condition that the corresponding utility and multiplicity values have to be encoded in unary required there. Concretely, we show that, while preserving fixed-parameter tractability, these values can be encoded in binary, thus greatly expanding the range of feasible values.Comment: 15 pages; Published in the Proceedings of ECAI-202

    Frobenius problem and the covering radius of a lattice

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    Let Nβ‰₯2N \geq2 and let 1<a1<...<aN1 < a_1 < ... < a_N be relatively prime integers. Frobenius number of this NN-tuple is defined to be the largest positive integer that cannot be expressed as βˆ‘i=1Naixi\sum_{i=1}^N a_i x_i where x1,...,xNx_1,...,x_N are non-negative integers. The condition that gcd(a1,...,aN)=1gcd(a_1,...,a_N)=1 implies that such number exists. The general problem of determining the Frobenius number given NN and a1,...,aNa_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 NN-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

    Integer Knapsacks: Average Behavior of the Frobenius Numbers

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    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

    Minimizing the number of lattice points in a translated polygon

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    The parametric lattice-point counting problem is as follows: Given an integer matrix A∈ZmΓ—nA \in Z^{m \times n}, compute an explicit formula parameterized by b∈Rmb \in R^m that determines the number of integer points in the polyhedron {x∈Rn:Ax≀b}\{x \in R^n : Ax \leq b\}. In the last decade, this counting problem has received considerable attention in the literature. Several variants of Barvinok's algorithm have been shown to solve this problem in polynomial time if the number nn of columns of AA is fixed. Central to our investigation is the following question: Can one also efficiently determine a parameter bb such that the number of integer points in {x∈Rn:Ax≀b}\{x \in R^n : Ax \leq b\} is minimized? Here, the parameter bb can be chosen from a given polyhedron QβŠ†RmQ \subseteq R^m. Our main result is a proof that finding such a minimizing parameter is NPNP-hard, even in dimension 2 and even if the parametrization reflects a translation of a 2-dimensional convex polygon. This result is established via a relationship of this problem to arithmetic progressions and simultaneous Diophantine approximation. On the positive side we show that in dimension 2 there exists a polynomial time algorithm for each fixed kk that either determines a minimizing translation or asserts that any translation contains at most 1+1/k1 + 1/k times the minimal number of lattice points
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