Approximating Subset Sum Ratio via Partition Computations

TL;DR

This paper introduces a new Fully Polynomial-Time Approximation Scheme (FPTAS) for the Subset Sum Ratio problem, leveraging advances in Partition problem algorithms to improve efficiency and reduce complexity compared to previous methods.

Contribution

The paper develops an FPTAS for Subset Sum Ratio that benefits from recent Partition problem improvements, reducing the complexity exponent and surpassing prior algorithms.

Findings

Improved complexity bounds for the FPTAS, with the exponent of n reduced to 2 in some cases.

The new scheme outperforms the previous best with complexity $O(n^4 / \varepsilon)$.

Established that the constant c in the complexity can be less than 5, improving theoretical bounds.

Abstract

We present a new FPTAS for the Subset Sum Ratio problem, which, given a set of integers, asks for two disjoint subsets such that the ratio of their sums is as close to $1$ as possible. Our scheme makes use of exact and approximate algorithms for the closely related Partition problem, hence any progress over those -- such as the recent improvement due to Bringmann and Nakos [SODA 2021] -- carries over to our FPTAS. Depending on the relationship between the size of the input set $n$ and the error margin $\varepsilon$, we improve upon the best currently known algorithm of Melissinos and Pagourtzis [COCOON 2018] of complexity $O(n^4 / \varepsilon)$. In particular, the exponent of $n$ in our proposed scheme may decrease down to $2$, depending on the Partition algorithm used. Furthermore, while the aforementioned state of the art complexity, expressed in the form $O((n + 1 / \varepsilon)^c)$, has constant $c = 5$, our results establish that $c < 5$.