In this paper, we address the online minimization knapsack problem, i. e., the items are given one by one over time and the goal is to minimize the total cost of items that covers a knapsack. We study the removable model, where it is allowed to remove old items from the knapsack in order to accept a new item. We obtain the following results. (i) We propose an 8-competitive deterministic algorit...

It is well known that the standard (linear) knapsack problem can be solved exactly by dynamic programming in O(nc) time, where n is the number of items and c is the capacity of the knapsack. The quadratic knapsack problem, on the other hand, is NP-hard in the strong sense, which makes it unlikely that it can be solved in pseudo-polynomial time. We show however that the dynamic programming appro...

We consider the average-case performance of a well-known approximation algorithm for the 0/1 knapsack problem, the Decreasing Density Greedy (DDG) algorithm. Let Un = {u1, . . . , un} be a set of n items, with each item ui having a size si and a profit pi, and Kn be the capacity of the knapsack. Given an instance of the 0/1 knapsack problem, let PL denote the total profit of an optimal solution...

We are concerned with a variation of the knapsack problem as well as of the knapsack sharing problem, where we are given a set of n items and a knapsack of a fixed capacity. As usual, each item is associated with its profit and weight, and the problem is to determine the subset of items to be packed into the knapsack. However, in the problem there are s players and the items are divided into s ...

We construct three public key knapsack cryptosystems. Standard knapsack cryptosystems hide easy instances of the knapsack problem and have been broken. The systems considered in the article face this problem: They hide a random (possibly hard) instance of the knapsack problem. We provide both complexity results (size of the key, time needed to encypher/decypher...) and experimental results. Sec...

In this note, we analyze a bilevel interdiction problem, where the follower’s program is a parametrized continuous knapsack. Based on the structure of the problem and an inverse optimization strategy, we propose for its solution an algorithm with worst-case complexity O(n). Keywords— Bilevel programming, Continuous knapsack problem, Polynomial time. 1 Model Recently, a number of papers have bee...

I cannot help but start this review with an analogy between the knapsack problem and the traveling salesman problem (TSP), the two most studied problems of combinatorial optimization. Both problems are easy to explain even to a stranger, both are hard to solve. Each of these problems is important in its own right as well as due to numerous applications. Each of the two problems forms an excelle...

Abstract We consider three new knapsack problems with variable weights or profits of items, where the weight profit an item depends on position in sequence items packed knapsack. show how to solve exactly using dynamic programming algorithms pseudo-polynomial running times and propose fully polynomial-time approximation schemes for their approximate solution.

We formulate the NP-hard n-dimensional knapsack feasibility problem as an equivalent absolute value equation (AVE) in an n-dimensional noninteger real variable space and propose a finite succession of linear programs for solving the AVE. Exact solutions are obtained for 1880 out of 2000 randomly generated consecutive knapsack feasibility problems with dimensions between 500 and one million. For...

In this paper, we address the online minimization knapsack problem, i. e., the items are given one by one over time and the goal is to minimize the total cost of items that covers a knapsack. We study the removable model, where it is allowed to remove old items from the knapsack in order to accept a new item. We obtain the following results. (i) We propose an 8-competitive deterministic algorit...