Simple and Faster Algorithms for Knapsack

In this paper, we obtain a number of new simple pseudo-polynomial time algorithms on the well-known knapsack problem, focusing on the running time dependency on the number of items $n$, the maximum item weight $w\mathrm{max}$, and the maximum item profit $p\mathrm{max}$. Our results include: - An $\widetilde{O}(n^{3/2}\cdot \min{w\mathrm{max},p\mathrm{max}})$-time randomized algorithm for 0-1 knapsack, improving the previous $\widetilde{O}(\min{n w\mathrm{max} p\mathrm{max}^{2/3},n p\mathrm{max} w\mathrm{max}^{2/3}})$ [Bringmann and Cassis, ESA'23] for the small $n$ case. - An $\widetilde{O}(n+\min{w\mathrm{max},p\mathrm{max}}^{5/2})$-time randomized algorithm for bounded knapsack, improving the previous $O(n+\min{w\mathrm{max}^3,p\mathrm{max}^3})$ [Polak, Rohwedder and Wegrzyck, ICALP'21].