• Journal of Korean Institute of Industrial Engineers
• /
• v.17 no.1
• /
• pp.127-130
• /
• 1991

The multi-divisional knapsack problem is defined as a binary knapsack problem where each of mutually exclusive divisions has its own capacity. We consider the relaxed LP problem and develop a transformation which converts the multi-divisional linear knapsack problem into smaller size linear knapsack problems. Solution procedures and a numerical example are presented.

• Journal of the Korea Institute of Information Security & Cryptology
• /
• v.1 no.1
• /
• pp.16-28
• /
• 1991

Knapsack public-key cryptosystems are based on the knapsack problem which is NP-complete. aii of the knapsack problem, are known to be insecure. However, the Chor and Rivest knapsack cryptosystem based on arithmetic in finite field is secure against all known cryptosystem based on arithmetic in a finite field is secure against all known cryptanalytic attacks. We suggest a new msthod of attack on knapsack cryptosystem which is based on the relaxation of a quadratic 0-1 integer optimization problem. We show that under certain condirions some bits of the solution of knapsack problem can be determined by using persistency property of linear relaxation. Also we propose a new Chor-Rivest system, this new cryptosystem reduces the number of calculation of discrete logarithms which are necessary for the implemention in a multi-user system.

• Journal of Korean Institute of Industrial Engineers
• /
• v.40 no.4
• /
• pp.396-403
• /
• 2014

The multi-divisional knapsack problem is defined as a binary knapsack problem where each mutually exclusive division has its own capacity. In this paper, we present an extension of the multi-divisional knapsack problem that has generalized multiple-choice constraints. We explore the linear programming relaxation (P) of this extended problem and identify some properties of problem (P). Then, we develop a transformation which converts the problem (P) into an LP knapsack problem and derive the optimal solutions of problem (P) from those of the converted LP knapsack problem. The solution procedures have a worst case computational complexity of order $O(n^2{\log}\;n)$, where n is the total number of variables. We illustrate a numerical example and discuss some variations of problem (P).

• Journal of Korean Institute of Industrial Engineers
• /
• v.27 no.1
• /
• pp.25-29
• /
• 2001

We present a two dimensional linear programming knapsack problem with the extended GUB constraint. The presented problem is an extension of the cardinality constrained linear programming knapsack problem. We identify some new properties of the problem and derive a solution algorithm based on the parametric analysis for the knapsack right-hand-side. The solution algorithm has a worst case time complexity of order O($n^2logn$). A numerical example is given.

• Management Science and Financial Engineering
• /
• v.14 no.1
• /
• pp.91-96
• /
• 2008

Robust knapsack problem appears when dealing with data uncertainty on the knapsack constraint. This note presents a generalization of the cover inequality for the problem with its lifting procedure. Specifically, we show that the lifting can be done in a polynomial time as in the usual knapsack problem. The results can serve as a building block in devising an efficient branch-and-cut algorithm for the general robust (0, 1) IP problem.

• Journal of the Korean Operations Research and Management Science Society
• /
• v.36 no.2
• /
• pp.43-50
• /
• 2011

We consider the separation problem of the rank-1 Chvatal-Gomory (C-G) inequalities for the 0-1 knapsack problem with the knapsack capacity defined by an additional binary variable, which we call the fixed-charge 0-1 knapsack problem. We analyze the structural properties of the optimal solutions to the separation problem and show that the separation problem can be solved in pseudo-polynomial time. By using the result, we also show that the existence of a pseudo-polynomial time algorithm for the separation problem of the rank-1 C-G inequalities of the ordinary 0-1 knapsack problem.

• Journal of the Korean Operations Research and Management Science Society
• /
• v.28 no.4
• /
• pp.191-198
• /
• 2003

The generalized knapsack problem or gknap is the combinatorial optimization problem of optimizing a nonnegative linear function over the integral hull of the intersection of a polynomially separable 0-1 polytope and a knapsack constraint. The knapsack, the restricted shortest path, and the constrained spanning tree problem are a partial list of gknap. More interesting1y, all the problem that are known to have a fully polynomial approximation scheme, or FPTAS are gknap. We establish some necessary and sufficient conditions for a gknap to admit an FPTAS. To do so, we recapture the standard scaling and approximate binary search techniques in the framework of gknap. This also enables us to find a weaker sufficient condition than the strong NP-hardness that a gknap does not have an FPTAS. Finally, we apply the conditions to explore the fully polynomial approximability of the constrained spanning problem whose fully polynomial approximability is still open.

• Proceedings of the Korean Operations and Management Science Society Conference
• /
• 2004.10a
• /
• pp.225-228
• /
• 2004

The generalized knapsack problem, or gknap is the combinatorial optimization problem of optimizing a nonnegative linear functional over the integral hull of the intersection of a polynomially separable 0 - 1 polytope and a knapsack constraint. Among many potential applications, the knapsack, the restricted shortest path, and the restricted spanning tree problem are such examples. We prove via the ellipsoid method the equivalence between the fully polynomial approximability and a certain pseudo-polynomial separability of the gknap polytope.

• Journal of Korean Institute of Industrial Engineers
• /
• v.38 no.2
• /
• pp.74-79
• /
• 2012

An efficient separation heuristic for the rank-1 Chvatal-Gomory cuts for the binary knapsack problem is proposed. The proposed heuristic is based on the decomposition property of the separation problem for the fixedcharge 0-1 knapsack problem characterized by Park and Lee [14]. Computational tests on the benchmark instances of the generalized assignment problem show that the proposed heuristic procedure can generate strong rank-1 C-G cuts more efficiently than the exact rank-1 C-G cut separation and the exact knapsack facet generation.

• Journal of Korean Society of Industrial and Systems Engineering
• /
• v.19 no.40
• /
• pp.137-142
• /
• 1996

An algorithm for solving the cardinality constrained linear programming knapsack problem is presented. The algorithm has a convenient structure for a branch-and-bound approach to the integer version, especially to the 0-1 collapsing knapsack problem. A numerical example is given.