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Abstract
We obtain a polynomial-time algorithm that, given input
( A , b ) , where
A
= ( B | N )
∈ ℤ m × n ,
m
<
n , with
nonsingular
B
∈ ℤ m × m
and
b
∈ ℤ m ,
finds a nonnegative integer solution to the system
A x
=
b
or determines that no such solution exists, provided that
b
is located sufficiently “deep” in the cone generated by the columns of
B . This
result improves on some of the previously known conditions that guarantee
polynomial-time solvability of linear Diophantine problems.
Keywords
multidimensional knapsack problem, polynomial-time
algorithms, asymptotic integer programming, lattice points,
Frobenius numbers
Mathematical Subject Classification 2010
Primary: 11D04, 90C10
Secondary: 11H06
Milestones
Received: 15 March 2019
Revised: 1 July 2019
Accepted: 15 July 2019
Published: 11 October 2019