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2019•arXiv (Cornell University)Open access

The Product Knapsack Problem: Approximation and Complexity

Ulrich Pferschy, Joachim Schauer, Clemens Thielen

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Abstract

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 the classical greedy procedure to the product knapsack problem.

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What this paper is about

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 the classical greedy procedure to the product knapsack problem.

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Available abstract

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 the classical greedy procedure to the product knapsack problem.

Key concepts: Knapsack problem, Polynomial-time approximation scheme, Continuous knapsack problem, Mathematics, Mathematical optimization, Product (mathematics), Greedy algorithm, Extension (predicate logic)

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