# An FPTAS for the Knapsack Problem with Parametric Weights

## TL;DR

This paper introduces the first fully polynomial-time approximation scheme (FPTAS) for the parametric weight knapsack problem, enabling efficient approximation across all parameter values with controlled accuracy.

## Contribution

The paper presents a novel FPTAS for the parametric weight knapsack problem, providing a solution that approximates optimal solutions for all parameter values efficiently.

## Key findings

- Achieves a running time of O(n^3/ε^2 * min{log^2 P, n^2} * min{log M, n log(n/ε)/log(n log(n/ε))})
- Provides (1-ε)-approximate solutions for all parameter values
- First FPTAS for this class of parametric knapsack problems

## Abstract

In this paper, we investigate the parametric weight knapsack problem, in which the item weights are affine functions of the form $w_i(\lambda) = a_i + \lambda \cdot b_i$ for $i \in \{1,\ldots,n\}$ depending on a real-valued parameter $\lambda$. The aim is to provide a solution for all values of the parameter. It is well-known that any exact algorithm for the problem may need to output an exponential number of knapsack solutions. We present the first fully polynomial-time approximation scheme (FPTAS) for the problem that, for any desired precision $\varepsilon \in (0,1)$, computes $(1-\varepsilon)$-approximate solutions for all values of the parameter. Our FPTAS is based on two different approaches and achieves a running time of $\mathcal{O}(n^3/\varepsilon^2 \cdot \min\{ \log^2 P, n^2 \} \cdot \min\{\log M, n \log (n/\varepsilon) / \log(n \log (n/\varepsilon) )\})$ where $P$ is an upper bound on the optimal profit and $M := \max\{W, n \cdot \max\{a_i,b_i: i \in \{1,\ldots,n\}\}\}$ for a knapsack with capacity $W$.