Simple and Faster Algorithms for Knapsack

TL;DR

This paper introduces new simple pseudo-polynomial algorithms for the knapsack problem, significantly improving running times by optimizing dependencies on item count, maximum weight, and maximum profit.

Contribution

It presents novel randomized algorithms for 0-1 and bounded knapsack with improved time complexities over previous methods.

Findings

Achieved an $ ilde{O}(n^{3/2} imes ext{min} ext{}igrace w_{max}, p_{max}igrace)$-time algorithm for 0-1 knapsack.

Developed an $ ilde{O}(n + ext{min} ext{}igrace w_{max}, p_{max}igrace)^{5/2}$-time algorithm for bounded knapsack.

Results improve upon previous algorithms in terms of efficiency and simplicity.

Abstract

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].