A Nearly Quadratic-Time FPTAS for Knapsack

TL;DR

This paper presents a nearly quadratic-time fully polynomial-time approximation scheme (FPTAS) for the Knapsack problem, improving the runtime significantly and establishing near-optimality under a common computational conjecture.

Contribution

The paper introduces a faster FPTAS for Knapsack with a runtime of O(n + (1/)^2), surpassing previous algorithms and showing near-optimality based on conjectured computational hardness.

Findings

Achieves O(n + (1/)^2) runtime for Knapsack FPTAS

Improves upon previous O(n + (1/)^{11/5}) algorithm

Establishes near-optimality conditioned on onjecture about onvolution complexity

Abstract

We investigate the classic Knapsack problem and propose a fully polynomial-time approximation scheme (FPTAS) that runs in $\widetilde{O}(n + (1/\varepsilon)^2)$ time. This improves upon the $\widetilde{O}(n + (1/\varepsilon)^{11/5})$-time algorithm by Deng, Jin, and Mao [\textit{Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms, 2023}]. Our algorithm is the best possible (up to a polylogarithmic factor) conditioned on the conjecture that $(\min, +)$-convolution has no truly subquadratic-time algorithm, since this conjecture implies that Knapsack has no $O((n + 1/\varepsilon)^{2-\delta})$-time FPTAS for any constant $\delta > 0$.