Possibility of Proving P = NP through the 3-Partition Problem

The P versus NP problem is a very intriguing concept as it asks whether difficult problems have an alternative easier solution, but the difficulty of a problem usually only exist from the perspective of the solver.

This paper details an alternative method to efficiently solve a 3-partition problem, which is considered a strongly NP-complete problem. Thus, proving that there exists an efficient method to solve a 3-partition problem might also prove P = NP.

Section 2 provides the logic behind the algorithm that efficiently solves a 3-partition problem. Section 3 provides proof of the algorithm’s efficiency by only taking 10 steps using an equation and a logic condition to find all possible subsets when given a set containing 27 elements. Section 4 provides additional proof by only taking 3 steps to find all possible subsets of a set containing 6 elements. Section 5 provides an overview of the algorithm used in this experiment and its limitations as a proof of concept.

Section 6 provides a detailed step by step application of the logic used by the algorithm using the set given on section 4. Section 7, 8, 9, and 10 details the solution implemented when the algorithm encounters special cases where multiple possible combinations exist. Section 11, 12, and 13 discusses the possibility of proving P = NP in the future through 3- partition problems. Additionally, section 14 provides a direct comparison of the results between a complete greedy algorithm and the proposed efficient algorithm.