Subset Sum in $O(n^{11}\log(n))$

TL;DR

This paper introduces the Dragonfly Algorithm, an approach to solve the subset-sum problem in polynomial time using a novel polynomial root intersection method.

Contribution

It presents a new polynomial-based algorithm with improved theoretical time complexity for the subset-sum problem.

Findings

Subset-sum can be solved in $O(n^{11}\log(n))$ time with the new algorithm.

The generalized product-derivative method efficiently generates systems of polynomials.

The approach leverages root intersections of polynomials to identify solutions.

Abstract

This paper describes an algorithm (thus far referred to as the "Dragonfly Algorithm") by which the subset-sum problem can be solved in $O(n^{11}\log(n))$ time complexity. The paper will first detail the generalized "product-derivative" method (and the more efficient version of this method which will be used in the algorithm) by which a pair of monic polynomials can be used to generate a system of unique monic polynomials for which each polynomial in the system will share with every other a set of roots equivalent to the intersection of the roots of the original pair; this method will then be applied on a pair of polynomials one of which, $\phi(x)$, exhibiting known roots based on the instance of the subset-sum problem and the other of which, $t(x)$, containing unknown placeholder coefficients and representing an unknown subset of the linear factors of $\phi(x)$.