0-1 Knapsack in Nearly Quadratic Time
Ce Jin
TL;DR
This paper advances the complexity frontier for 0-1 Knapsack in the pseudopolynomial regime by presenting a deterministic $O(n + w_{\max}^{2}\log^{4} w_{\max})$-time algorithm that matches the $(n+w_{\max})^{2-o(1)}$ conditional lower bounds up to subpolynomial factors. It fuses fine-grained proximity, witness propagation adapted to the 0-1 setting, and a two-stage weight-partitioned DP with rank-based structure, enabling a near-quadratic dependence on $w_{\max}$ while keeping $n$ additive. Central to the approach are the refined DP techniques via hints (\textsc{HintedKnapsackExtend}^+), the SMAWK-based batch updates for convex Monge-like structures, and color-coding to manage larger hint sets, all orchestrated to preserve optimality through careful exchange arguments. The results close the gap between fastest known algorithms and conditional lower bounds, with corollaries for $p_{\max}$, and connect to independent concurrent work; the methods also suggest directions for Subset Sum and related knapsack variants. Overall, the work introduces a robust toolkit that blends additive combinatorics, proximity analyses, and derandomized color-coding to achieve near-quadratic running times for a foundational NP-hard problem in the pseudopolynomial setting.
Abstract
We study pseudo-polynomial time algorithms for the fundamental \emph{0-1 Knapsack} problem. Recent research interest has focused on its fine-grained complexity with respect to the number of items $n$ and the \emph{maximum item weight} $w_{\max}$. Under $(\min,+)$-convolution hypothesis, 0-1 Knapsack does not have $O((n+w_{\max})^{2-δ})$ time algorithms (Cygan-Mucha-Węgrzycki-Włodarczyk 2017 and Künnemann-Paturi-Schneider 2017). On the upper bound side, currently the fastest algorithm runs in $\tilde O(n + w_{\max}^{12/5})$ time (Chen, Lian, Mao, and Zhang 2023), improving the earlier $O(n + w_{\max}^3)$-time algorithm by Polak, Rohwedder, and Węgrzycki (2021). In this paper, we close this gap between the upper bound and the conditional lower bound (up to subpolynomial factors): - The 0-1 Knapsack problem has a deterministic algorithm in $O(n + w_{\max}^{2}\log^4w_{\max})$ time. Our algorithm combines and extends several recent structural results and algorithmic techniques from the literature on knapsack-type problems: - We generalize the "fine-grained proximity" technique of Chen, Lian, Mao, and Zhang (2023) derived from the additive-combinatorial results of Bringmann and Wellnitz (2021) on dense subset sums. This allows us to bound the support size of the useful partial solutions in the dynamic program. - To exploit the small support size, our main technical component is a vast extension of the "witness propagation" method, originally designed by Deng, Mao, and Zhong (2023) for speeding up dynamic programming in the easier unbounded knapsack settings. To extend this approach to our 0-1 setting, we use a novel pruning method, as well as the two-level color-coding of Bringmann (2017) and the SMAWK algorithm on tall matrices.