We consider a knapsack problem with precedence constraints imposed on pairs of items, known as the precedence constrained knapsack problem (PCKP). This problem has applications in manufacturing and mining, and also appears as a subproblem in decomposition techniques for network design and related problems. We present a new approach for determining facets of the PCKP polyhedron based on clique i...

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Recently, Cygan, Kowalik, and Wykurz [IPL 2009] gave sub-exponential-time approximation algorithms for the Set Cover problem with approximation ratios better than lnn. In light of this result, it is natural to ask whether such improvements can be achieved using lift-andproject methods. We present a simpler combinatorial algorithm which has nearly the same time-approximation tradeoff as the algo...

2 Knapsack Problem (continued) 2.1 Algorithm Recap Definition: We define states (sub-problems) as the following: the maximum value for a knapsack with capacity j and we are given the first i items. Then, we have ∀i, j, the corresponding state, a[i, j] is max(a[i−1, j−wi] + vi, a[i−1, j]), where the first option is the value if item i is in the knapsack and the second option is the value if item...

The 0/1 knapsack problem is a well known problem occurring in many real world problems. The problem is NP-Complete. The multiobjective 0/1 knapsack problem is a generalization of the 0/1 knapsack problem in which multiple knapsacks are considered. A new evolutionary algorithm for solving multiobjective 0/1 knapsack problem is proposed in this paper. This algorithm used a ε-dominance relation fo...

It is well known that the knapsack cryptosystem is a type of public key cryptosystems. It is based on the well known mathematical concept, i.e., the choosing of one way function. Many Knapsack cryptosystems were invented in the last 20 years, and many attacking methods were invented as a counterattack against the developed knapsack cryptosystems. All the attacking methods aim towards breaking t...

Ralph Gomory has recently aroused interest in a special type of knapsack problem in which the constraint coefficients and constant term are elements of a finite additive group. The significance of this problem lies in the fact that it is closely related to the general integer linear programming problem, resulting by removing the nonnegativity restrictions on those variables in the general probl...

We consider the polyhedral structure of the precedence constrained knapsack (PCK) problem, also known as the partially ordered knapsack problem. A set of items N is given, along with a partial order, or set of precedence relationships, on the items, denoted by S ⊆ N × N. A precedence relationship (i, j) ∈ S exists if item i can be placed in the knapsack only if item j is in the knapsack. Each i...