On the existence of polynomial-time algorithms to the subset sum problem

TL;DR

This paper demonstrates that the subset sum problem cannot be solved in polynomial time, implying that P does not equal NP, which has significant implications for computational complexity theory.

Contribution

It provides a proof that no polynomial-time algorithm exists for the subset sum problem, establishing a key separation in complexity classes.

Findings

No polynomial-time algorithm for subset sum exists

P does not equal NP based on this problem

Implications for computational complexity theory

Abstract

This paper proves that there does not exist a polynomial-time algorithm to the the subset sum problem. As this problem is in NP, the result implies that the class P of problems admitting polynomial-time algorithms does not equal the class NP of problems admitting nondeterministic polynomial-time algorithms.