TY - GEN
T1 - Hard equality constrained integer knapsacks
AU - Aardal, K.I.
AU - Lenstra, A.K.
PY - 2002
Y1 - 2002
N2 - We consider the following integer feasibility problem: "Given positive integer numbers a 0, a 1,..., a n, with gcd(a 1,..., a n) = 1 and a = (a 1,..., a n), does there exist a nonnegative integer vector x satisfying ax = a 0?" Some instances of this type have been found to be extremely hard to solve by standard methods such as branch-and-bound, even if the number of variables is as small as ten. We observe that not only the sizes of the numbers a 0, a 1,..., a n, but also their structure, have a large impact on the difficulty of the instances. Moreover, we demonstrate that the characteristics that make the instances so difficult to solve by branch-and-bound make the solution of a certain reformulation of the problem almost trivial. We accompany our results by a small computational study.
AB - We consider the following integer feasibility problem: "Given positive integer numbers a 0, a 1,..., a n, with gcd(a 1,..., a n) = 1 and a = (a 1,..., a n), does there exist a nonnegative integer vector x satisfying ax = a 0?" Some instances of this type have been found to be extremely hard to solve by standard methods such as branch-and-bound, even if the number of variables is as small as ten. We observe that not only the sizes of the numbers a 0, a 1,..., a n, but also their structure, have a large impact on the difficulty of the instances. Moreover, we demonstrate that the characteristics that make the instances so difficult to solve by branch-and-bound make the solution of a certain reformulation of the problem almost trivial. We accompany our results by a small computational study.
U2 - 10.1007/3-540-47867-1_25
DO - 10.1007/3-540-47867-1_25
M3 - Conference contribution
SN - 3-540-43676-6
T3 - Lecture Notes in Computer Science
SP - 350
EP - 366
BT - Proceedings 9th IPCO (Cambridge MA, USA, May 27-29, 2002)
A2 - Cook, W.J.
A2 - Schulz, A.S.
PB - Springer
CY - Berlin
ER -