We study pseudo-polynomial time algorithms for the fundamental 0-1 Knapsack problem. Recent research interest has focused on its finegrained complexity with respect to the number of items.. and the maximum item weight w(max). Under (min, +)-convolution hypothesis, 0-1 Knapsack does not have O((n +w(max))(2-sigma)) time algorithms (Cygan-Mucha-Wegrzycki-Wlodarczyk 2017 and Kunnemann-PaturiSchneider 2017). On the upper bound side, currently the fastest algorithm runs in 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 Wegrzycki (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(4.)w(max)) time. Our algorithm combines and extends several recent structural results and algorithmic techniques from the literature on knapsacktype problems: (1) We generalize the "fine-grained proximity" technique of Chen, Lian, Mao, and Zhang (2023) derived from the additivecombinatorial 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. (2) 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.
