Journal of Symbolic Computation
Journal of Symbolic Computation
Integer Knapsacks: Average Behavior of the Frobenius Numbers
Mathematics of Operations Research
Basis reduction and the complexity of branch-and-bound
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Journal of Global Optimization
Feasibility of Integer Knapsacks
SIAM Journal on Optimization
Bounds on the size of branch-and-bound proofs for integer knapsacks
Operations Research Letters
Column basis reduction and decomposable knapsack problems
Discrete Optimization
LLL-reduction for integer knapsacks
Journal of Combinatorial Optimization
On the structure of reduced kernel lattice bases
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
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We consider the following integer feasibility problem: Given positive integer numbersa0,a1, ... , a n , with gcd( a1,..., a n ) = 1 anda = ( a1,..., a n ), does there exist a vectorx?Z n =0satisfyinga x= a0? We prove that if the coefficientsa1,..., a nhave a certain decomposable structure, then the Frobenius number associated witha1,..., a n , i.e., the largest value ofa0 for whicha x= a0 does not have a nonnegative integer solution, is close to a known upper bound. In the instances we consider, we takea0 to be the Frobenius number. Furthermore, we show that the decomposable structure ofa1,..., a nmakes the solution of a lattice reformulation of our problem almost trivial, since the number of lattice hyperplanes that intersect the polytope resulting from the reformulation in the direction of the last coordinate is going to be very small. For branch-and-bound such instances are difficult to solve, since they are infeasible and have large values ofa0/ a i , 1= i= n. We illustrate our results by some computational examples.