A Nearly Quadratic-Time FPTAS for Knapsack

TL;DR

This work presents a fully polynomial-time approximation scheme for Knapsack with running time $\\tilde{O}(n + (1/\\varepsilon)^2)$, outperforming the previous $\\tilde{O}(n + (1/\\varepsilon)^{11/5})$ bound. The authors fuse proximity results—showing that optimal solutions largely use high-efficiency items—with a functional approximation framework that computes $f_I(x)$ for all capacities via $\\max, +$-convolution and grouping by profits. A carefully designed reduction RP$(\\varepsilon, \\alpha)$, together with tail/head decomposition and additive-combinatorics-based proximity bounds, yields an efficient algorithm that achieves the target time up to polylogarithmic factors, conditional on hardness of $\\min, +$-convolution. The result advances the state of the art for Knapsack, supports near-quadratic-time FPTAS lower bounds under a standard conjecture, and motivates further exploration of the boundary between exact, approximate, and weakly-approximate regimes in combinatorial optimization.

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 $O((n + 1/\varepsilon)^{2-δ})$-time FPTAS for any constant $δ> 0$.

Figures (3)

• Figure 1: $I_H$ is superior to $b$ and $I_L$ is inferior to $b$ by $\tau$ distnct profits
• Figure 2: Partition of $I_{\mathrm{tail}}$
• Figure 3: A Partition of $I_{\mathrm{head}}$ for $[t_j, t_{j+1}]$

Theorems & Definitions (45)

• Theorem 1
• Lemma 2
• proof
• Lemma 3: Chan18
• Lemma 4
• Lemma 5
• Lemma 6
• proof : Proof of Lemma \ref{['lem:proximity']}
• Definition 7: BW21
• Definition 8: BW21