Fully Subexponential Time Approximation Scheme for Product Partition

TL;DR

This paper introduces a subexponential time approximation scheme for the Product Partition Problem by leveraging integer factorization and prime exponent matrices to efficiently find solutions.

Contribution

The paper presents a novel subexponential algorithm for PPP that uses prime factorization and matrix modifications to improve solution search efficiency.

Findings

Algorithm runs in subexponential time and memory.

Effective factorization and matrix techniques facilitate solution finding.

Approach maintains product integrity while enabling efficient search.

Abstract

In this paper we study the Product Partition Problem (PPP), i.e. we are given a set of $n$ natural numbers represented on $m$ bits each and we are asked if a subset exists such that the product of the numbers in the subset equals the product of the numbers not in the subset. Our approach is to obtain the integer factorization of each number. This is the subexponential step. We then form a matrix with the exponents of the primes and propose a novel procedure which modifies the given numbers in such a way that their integer factorization contains sufficient primes to facilitate the search for the solution to the partition problem, while maintaining a similar product. We show that the required time and memory to run the proposed algorithm is subexponential.