Approximating Knapsack and Partition via Dense Subset Sums

TL;DR

This paper presents improved algorithms for approximating Knapsack and Partition problems, achieving faster runtimes by leveraging dense subset sum techniques and novel divide-and-conquer methods, advancing the understanding of their computational complexity.

Contribution

The paper introduces the first application of dense subset sum techniques to Knapsack and Partition, improving approximation algorithms and runtime bounds.

Findings

Knapsack approximation time improved to $ ilde O(n + (1/\varepsilon)^{2.2})$

Partition approximation time improved to $ ilde O(n + (1/\varepsilon)^{1.25})$

New divide-and-conquer methods speed up additive problem solutions

Abstract

Knapsack and Partition are two important additive problems whose fine-grained complexities in the $(1-\varepsilon)$-approximation setting are not yet settled. In this work, we make progress on both problems by giving improved algorithms. - Knapsack can be $(1 - \varepsilon)$-approximated in $\tilde O(n + (1/\varepsilon) ^ {2.2} )$ time, improving the previous $\tilde O(n + (1/\varepsilon) ^ {2.25} )$ by Jin (ICALP'19). There is a known conditional lower bound of $(n+\varepsilon)^{2-o(1)}$ based on $(\min,+)$-convolution hypothesis. - Partition can be $(1 - \varepsilon)$-approximated in $\tilde O(n + (1/\varepsilon) ^ {1.25} )$ time, improving the previous $\tilde O(n + (1/\varepsilon) ^ {1.5} )$ by Bringmann and Nakos (SODA'21). There is a known conditional lower bound of $(1/\varepsilon)^{1-o(1)}$ based on Strong Exponential Time Hypothesis. Both of our new algorithms apply the additive combinatorial results on dense subset sums by Galil and Margalit (SICOMP'91), Bringmann and Wellnitz (SODA'21). Such techniques have not been explored in the context of Knapsack prior to our work. In addition, we design several new methods to speed up the divide-and-conquer steps which naturally arise in solving additive problems.