Fully Subexponential Time Approximation Scheme for Product Partition
Marius Costandin
TL;DR
This work tackles the Product Partition Problem (PPP) by targeting a subexponential-time approach that first factors the input numbers and then encodes their prime exponents in a matrix $S_M$, converting PPP into a linear-algebraic condition on $x \in \{0,1\}^n$. To enhance tractability, it introduces an Unambiguous PPP (UPPP) framework and a Primes Pump Algorithm that enlarges the prime basis via a $\gamma$-pumping scheme, aiming to satisfy $n \le q$ and enable a subexponential search while keeping the product close. The core idea reduces PPP to solving multiple subset-sum-like constraints and leverages eigenstructure to guide approximate solutions when exact solutions are not unique. Overall, the paper sketches a Fully Subexponential Time Approximation Scheme (FSTAS) for PPP, relying on richer factorizations and controlled transformations to achieve subexponential time and memory growth with potential implications for NP-hard partition-type problems.
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.