71 research outputs found
An Optimal Lower Bound for the Frobenius Problem
Given positive integers with , let
denote the largest natural number which is not a positive integer
combination of . This paper gives an optimal lower bound for
in terms of the absolute inhomogeneous minimum of the standard
-simplex.Comment: 10 page
Normal Toric Ideals of Low Codimension
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
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
Let and let be relatively prime integers.
Frobenius number of this -tuple is defined to be the largest positive
integer that cannot be expressed as where
are non-negative integers. The condition that implies that
such number exists. The general problem of determining the Frobenius number
given and 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 -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
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
The parametric lattice-point counting problem is as follows: Given an integer
matrix , compute an explicit formula parameterized by that determines the number of integer points in the polyhedron . 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 of columns of is fixed.
Central to our investigation is the following question: Can one also
efficiently determine a parameter such that the number of integer points in
is minimized? Here, the parameter can be chosen
from a given polyhedron .
Our main result is a proof that finding such a minimizing parameter is
-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 that either determines a minimizing
translation or asserts that any translation contains at most times
the minimal number of lattice points
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