[Paper Review] The Product Knapsack Problem: Approximation and Complexity

Ulrich Pferschy, Joachim Schauer+1 more|arXiv (Cornell University)|Jan 3, 2019

Optimization and Search Problems3 references3 citations

TL;DR

This paper establishes the product knapsack problem (PKP) as weakly NP-hard and presents a fully polynomial-time approximation scheme (FPTAS) based on dynamic programming in the profits space. It also analyzes a natural greedy extension, showing it achieves only a suboptimal approximation ratio, and proves the FPTAS is optimal under standard complexity assumptions.

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.

Motivation & Objective

• To establish the computational complexity of the product knapsack problem (PKP), where the objective is to maximize the product of selected item profits.
• To develop a fully polynomial-time approximation scheme (FPTAS) for PKP, overcoming the failure of standard greedy algorithms to achieve constant approximation ratios.
• To analyze the approximation quality of a natural extension of the classical greedy algorithm to the product knapsack setting.
• To prove that the proposed FPTAS achieves optimal approximation performance under standard complexity assumptions.
• To demonstrate that the greedy approach for PKP yields only a suboptimal approximation ratio, and that this bound is asymptotically tight.

Proposed method

• Prove weak NP-hardness of PKP via a reduction from the classical 0-1 knapsack problem, using a transformation that preserves solution structure.
• Construct an FPTAS using dynamic programming based on the logarithmic transformation of profits, reducing the product maximization to a sum maximization problem.
• Use a splitting technique for items with negative profits, introducing a split item to handle parity constraints in the number of negative-profit items.
• Apply a rounding procedure on logarithmic values of profits to achieve polynomial-time approximation with controlled error bounds.
• Analyze the approximation ratio by bounding the logarithmic objective value of the heuristic solution relative to the optimal solution in logarithmic space.
• Establish tightness of the approximation ratio by constructing an example where the greedy algorithm’s performance matches the theoretical bound.

Experimental results

Research questions

• RQ1 Is the product knapsack problem (PKP) weakly NP-hard, and can its complexity be formally established?
• RQ2 Can a fully polynomial-time approximation scheme (FPTAS) be constructed for PKP despite the failure of classical greedy heuristics?
• RQ3 What is the best possible approximation ratio achievable by a natural extension of the greedy algorithm to PKP?
• RQ4 How does the performance of the greedy algorithm compare to the optimal solution in the worst case?
• RQ5 Can the FPTAS be shown to be optimal in terms of approximation ratio under standard complexity assumptions?

Key findings

• The product knapsack problem is weakly NP-hard, as proven via a reduction from the classical 0-1 knapsack problem.
• An FPTAS for PKP is constructed using dynamic programming on the logarithmic values of profits, achieving a (1−ε)-approximation in time polynomial in n, C, and 1/ε.
• The greedy algorithm extended to PKP achieves an approximation ratio of at best (z*)^{1/3}, and this bound is asymptotically tight.
• The additive error in the logarithmic space of the greedy solution is bounded by 2·log(p_max), leading to a worst-case approximation ratio of z*^{1/3} · p_max^2.
• The FPTAS construction is optimal in the sense that no better approximation ratio can be achieved in polynomial time unless P=NP.
• An example is provided where the greedy algorithm's solution value is only 2(M+2), while the optimal solution achieves (M+2)M², confirming the theoretical approximation bound.