Solving Knapsack with Small Items via L0-Proximity

We study pseudo-polynomial time algorithms for the fundamental \emph{0-1 Knapsack} problem. In terms of $n$ and $w{\max}$, previous algorithms for 0-1 Knapsack have cubic time complexities: $O(n^2w{\max})$ (Bellman 1957), $O(nw{\max}^2)$ (Kellerer and Pferschy 2004), and $O(n + w{\max}^3)$ (Polak, Rohwedder, and W\k{e}grzycki 2021). On the other hand, fine-grained complexity only rules out $O((n+w{\max})^{2-\delta})$ running time, and it is an important question in this area whether $\tilde O(n+w{\max}^2)$ time is achievable. Our main result makes significant progress towards solving this question: - The 0-1 Knapsack problem has a deterministic algorithm in $\tilde O(n + w{\max}^{2.5})$ time. Our techniques also apply to the easier \emph{Subset Sum} problem: - The Subset Sum problem has a randomized algorithm in $\tilde O(n + w{\max}^{1.5})$ time. This improves (and simplifies) the previous $\tilde O(n + w{\max}^{5/3})$-time algorithm by Polak, Rohwedder, and W\k{e}grzycki (2021) (based on Galil and Margalit (1991), and Bringmann and Wellnitz (2021)). Similar to recent works on Knapsack (and integer programs in general), our algorithms also utilize the \emph{proximity} between optimal integral solutions and fractional solutions. Our new ideas are as follows: - Previous works used an $O(w{\max})$ proximity bound in the $\ell1$-norm. As our main conceptual contribution, we use an additive-combinatorial theorem by Erd\H{o}s and S\'{a}rk\"{o}zy (1990) to derive an $\ell0$-proximity bound of $\tilde O(\sqrt{w{\max}})$. - Then, the main technical component of our Knapsack result is a dynamic programming algorithm that exploits both $\ell0$- and $\ell_1$-proximity. It is based on a vast extension of the ``witness propagation'' method, originally designed by Deng, Mao, and Zhong (2023) for the easier \emph{unbounded} setting only.