A Subexponential Reduction from Product Partition to Subset Sum

TL;DR

This paper presents a subexponential reduction from the Product Partition Problem to a single Subset Sum Problem by leveraging integer factorization and prime exponent matrices, simplifying the solution process.

Contribution

It introduces a novel reduction method from PPP to SSP, utilizing prime factorization and exponent matrices to unify multiple subset sum problems into one polynomial-sized problem.

Findings

PPP solution iff the final SSP has a solution

Reduction is subexponential in the size of input numbers

Unified SSP is polynomial in input size

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 show that the PPP has a solution iff some Subset Sum Problems have a common solution. Finally, using the fact that the exponents are not large we combine all the Subset Sum Problems in a single Subset Sum Problem (SSP) and show that its size is polynomial in $m,n$. We show that the PPP has a solution iff the final SSP has one.