A Nearly Quadratic-Time FPTAS for Knapsack

2023-08-15

Symposium on the Theory of Computing

AI Summary

This paper addresses the classic 0–1 knapsack problem and presents the first fully polynomial-time approximation scheme (FPTAS) achieving a tight time complexity. Methodologically, it integrates lower-bound analysis for (min, +)-convolution, fine-grained dynamic programming pruning, and piecewise approximation techniques. The resulting algorithm runs in $O(n + (1/varepsilon)^2)$ time—improving upon the previous best bound of $O(n + (1/varepsilon)^{11/5})$ and matching the theoretical lower bound up to polylogarithmic factors. Under standard computational assumptions, this complexity is provably optimal and cannot be further improved. Consequently, this work establishes the tight computational complexity threshold for FPTASes for the 0–1 knapsack problem, providing both a new theoretical benchmark and a novel technical paradigm for the design of approximation algorithms.

Abstract

We investigate the classic Knapsack problem and propose a fully polynomial-time approximation scheme (FPTAS) that runs in O(n + (1/)2) time. Prior to our work, the best running time is O(n + (1/)11/5) [Deng, Jin, and Mao’23]. Our algorithm is the best possible (up to a polylogarithmic factor), as Knapsack has no O((n + 1/)2−δ)-time FPTAS for any constant δ > 0, conditioned on the conjecture that (min, +)-convolution has no truly subquadratic-time algorithm.

Problem

• Knapsack Problem
• Algorithm Design
• Quadratic Time Complexity

Innovation

• Knapsack Problem
• Improved Time Complexity
• Optimal Algorithm