A Critique of Quigley's "A Polynomial Time Algorithm for 3SAT"

TL;DR

This paper critically analyzes Quigley's claimed polynomial-time algorithm for 3SAT, demonstrating its flaws and providing counterexamples that invalidate its correctness, thereby refuting the claim that P equals NP based on this algorithm.

Contribution

The paper identifies errors in Quigley's proof and constructs counterexamples, showing his algorithm does not solve 3SAT in polynomial time.

Findings

Quigley's algorithm fails on certain inputs

Counterexamples show the algorithm's incorrect classification

The claim that P=NP is not supported by Quigley's work

Abstract

In this paper, we examine Quigley's "A Polynomial Time Algorithm for 3SAT" [Qui24]. Quigley claims to construct an algorithm that runs in polynomial time and determines whether a boolean formula in 3CNF form is satisfiable. Such a result would prove that 3SAT $\in \text{P}$ and thus $\text{P} = \text{NP}$. We show Quigley's argument is flawed by providing counterexamples to several lemmas he attempts to use to justify the correctness of his algorithm. We also provide an infinite class of 3CNF formulas that are unsatisfiable but are classified as satisfiable by Quigley's algorithm. In doing so, we prove that Quigley's algorithm fails on certain inputs, and thus his claim that $\text{P} = \text{NP}$ is not established by his paper.