A Faster FPTAS for the Subset-Sums Ratio Problem

TL;DR

This paper introduces a significantly faster Fully Polynomial-Time Approximation Scheme (FPTAS) for the Subset-Sums Ratio problem, improving computational efficiency by innovative dynamic programming techniques.

Contribution

The paper presents a new FPTAS that uses a dynamic programming table with a difference-based dimension, enhancing speed over previous methods.

Findings

FPTAS is several orders of magnitude faster than previous schemes.

The new approach effectively approximates the Subset-Sums Ratio problem.

Use of a difference-based dynamic programming table improves efficiency.

Abstract

The Subset-Sums Ratio problem (SSR) is an optimization problem in which, given a set of integers, the goal is to find two subsets such that the ratio of their sums is as close to 1 as possible. In this paper we develop a new FPTAS for the SSR problem which builds on techniques proposed in [D. Nanongkai, Simple FPTAS for the subset-sums ratio problem, Inf. Proc. Lett. 113 (2013)]. One of the key improvements of our scheme is the use of a dynamic programming table in which one dimension represents the difference of the sums of the two subsets. This idea, together with a careful choice of a scaling parameter, yields an FPTAS that is several orders of magnitude faster than the best currently known scheme of [C. Bazgan, M. Santha, Z. Tuza, Efficient approximation algorithms for the Subset-Sums Equality problem, J. Comp. System Sci. 64 (2) (2002)].