منابع مشابه
Feasibility of Integer Knapsacks
FEASIBILITY OF INTEGER KNAPSACKS∗ ISKANDER ALIEV† AND MARTIN HENK‡ Abstract. Given a matrix A ∈ Zm×n satisfying certain regularity assumptions, we consider the set F(A) of all vectors b ∈ Zm such that the associated knapsack polytope P (A, b) = {x ∈ R≥0 : Ax = b} contains an integer point. When m = 1 the set F(A) is known to contain all consecutive integers greater than the Frobenius number ass...
متن کاملHard Equality Constrained Integer Knapsacks
We consider the following integer feasibility problem: Given positive integer numbers a0 a1 an, with gcd a1 an = 1 and a = a1 an , does there exist a vector x ∈ n≥0 satisfying a x = a0? We prove that if the coefficients a1 an have a certain decomposable structure, then the Frobenius number associated with a1 an, i.e., the largest value of a0 for which a x= a0 does not have a nonnegative integer...
متن کاملLLL-reduction for integer knapsacks
Given a matrix A ∈ Zm×n satisfying certain regularity assumptions, a wellknown integer programming problem asks to find an integer point in the associated knapsack polytope P(A,b)= {x ∈R≥0 :Ax = b} or determine that no such point exists. We obtain an LLL-based polynomial time algorithm that solves the problem subject to a constraint on the location of the vector b.
متن کاملLattice Reduction, Integer Programming, and Knapsacks
We will discuss knapsack problems that arise in certain computa− tional number theory settings. A common theme is that the search space for the standard real relaxation is large; in a sense this trans− lates to a poor choice of variables. Lattice reduction methods have been developed in the past few years to improve handling of such problems. We show explicitly how they may be applied to comput...
متن کاملInteger Knapsacks: Average Behavior of the Frobenius Numbers
Given a primitive integer vector a ∈ Z>0, the largest integer b such that the knapsack polytope P = {x ∈ R≥0 : 〈a,x〉 = b} contains no integer point is called the Frobenius number of a. We show that the asymptotic growth of the Frobenius number in average is significantly slower than the growth of the maximum Frobenius number. More precisely, we prove that it does not essentially exceed ||a|| 1+...
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ژورنال
عنوان ژورنال: SIAM Journal on Optimization
سال: 2010
ISSN: 1052-6234,1095-7189
DOI: 10.1137/090778043