# An Improved FPTAS for 0-1 Knapsack

## TL;DR

This paper presents an improved fully polynomial-time approximation scheme (FPTAS) for the 0-1 knapsack problem, reducing the runtime closer to the theoretical lower bound through advanced algorithmic techniques.

## Contribution

The paper introduces a faster FPTAS for 0-1 knapsack using a multi-level number-theoretic approach and a greedy lemma to minimize unnecessary computations.

## Key findings

- Runtime improved to  O(n+(1/)^{9/4})
- Approaches the quadratic lower bound more closely
- Enhances efficiency over previous FPTAS by Chan (2018)

## Abstract

The 0-1 knapsack problem is an important NP-hard problem that admits fully polynomial-time approximation schemes (FPTASs). Previously the fastest FPTAS by Chan (2018) with approximation factor $1+\varepsilon$ runs in $\tilde O(n + (1/\varepsilon)^{12/5})$ time, where $\tilde O$ hides polylogarithmic factors. In this paper we present an improved algorithm in $\tilde O(n+(1/\varepsilon)^{9/4})$ time, with only a $(1/\varepsilon)^{1/4}$ gap from the quadratic conditional lower bound based on $(\min,+)$-convolution. Our improvement comes from a multi-level extension of Chan's number-theoretic construction, and a greedy lemma that reduces unnecessary computation spent on cheap items.