# The Product Knapsack Problem: Approximation and Complexity

@article{Pferschy2019ThePK, title={The Product Knapsack Problem: Approximation and Complexity}, author={Ulrich Pferschy and Joachim Schauer and Clemens Thielen}, journal={ArXiv}, year={2019}, volume={abs/1901.00695} }

We consider the product knapsack problem, which is the variant of the classical 0-1 knapsack problem where the objective consists of maximizing the product of the profits of the selected items. These profits are allowed to be positive or negative. We show that this recently introduced variant of the knapsack problem is weakly NP-hard and present a fully polynomial-time approximation scheme (FPTAS) for the problem. Moreover, we analyze the approximation quality achieved by a natural extension of…

## Tables from this paper

## 3 Citations

Approximating the product knapsack problem

- Mathematics, Computer ScienceOptim. Lett.
- 2021

This work presents the first fully polynomial-time approximation scheme for the product knapsack problem, which is known to be weakly -hard, and analyzes the approximation quality achieved by a natural extension of the classicalknapsack greedy procedure to the product Knapsack Problem.

A hybrid grey wolf optimizer for solving the product knapsack problem

- Computer ScienceInt. J. Mach. Learn. Cybern.
- 2021

Comparison results show that ABGWO is superior to others in terms of solution quality, robustness and convergence speed, and it is most suitable for solving PKP.

Functional Uncertainty in Real-Time Safety-Critical Systems

- Computer ScienceRTNS
- 2022

A model is proposed in which components are characterized by bounds on the degree of uncertainty under both worst-case and typical circumstances; the objective is to assure safety under all circumstances while optimizing for performance for typical circumstances.

## References

SHOWING 1-7 OF 7 REFERENCES

O C ] 3 J an 2 01 9 The Product Knapsack Problem : Approximation and Complexity

- Mathematics, Computer Science
- 2019

It is shown that this recently introduced variant of the knapsack problem is weakly NP-hard and present a fully polynomial-time approximation scheme (FPTAS) for the problem.

Knapsack Problems: Algorithms and Computer Implementations

- Computer Science, Mathematics
- 1990

This paper focuses on the part of the knapsack problem where the problem of bin packing is concerned and investigates the role of computer codes in the solution of this problem.

"Product Partition" and related problems of scheduling and systems reliability: Computational complexity and approximation

- BusinessEur. J. Oper. Res.
- 2010

Bi-criteria path problem with minimum length and maximum survival probability

- Mathematics, Computer ScienceOR Spectr.
- 2019

It is proved that the problem of finding a path which satisfies two bounds, one for each criterion, is NP-complete, even in the acyclic case.

On the Product Knapsack Problem

- Computer ScienceOptim. Lett.
- 2018

It is shown that PKP is weakly NP-hard, and a Dynamic Programming algorithm and different Mixed Integer Linear and Nonlinear Programming formulations for the PKP are proposed and implemented.

C

- Thielen, The product knapsack problem: Approximation and complexity
- 2019