On Salum's Algorithm for $\mathrm{X3SAT}$

TL;DR

This paper critiques Salum's claimed polynomial-time algorithm for X3SAT, demonstrating that its lack of backtracking leads to incorrect solutions, thus challenging the claim that P equals NP.

Contribution

It provides a counterexample showing the flaw in Salum's non-backtracking algorithm for X3SAT, questioning its validity and the broader claim of P=NP.

Findings

Counterexample invalidates Salum's algorithm

Lack of backtracking causes incorrect results

Challenges the claim that P=NP

Abstract

This is a commentary on, and critique of, Latif Salum's paper titled "Tractability of One-in-three $\mathrm{3SAT}$: $\mathrm{P} = \mathrm{NP}$." Salum purports to give a polynomial-time algorithm that solves the $\mathrm{NP}$-complete problem $\mathrm{X3SAT}$, thereby claiming $\mathrm{P} = \mathrm{NP}$. The algorithm, in short, fixes the polarity of a variable, carries out simplifications over the resulting formula to decide whether to keep the value assigned or flip the polarity, and repeats with the remaining variables. One thing this algorithm does not do is backtrack. We give an illustrative counterexample showing why the lack of backtracking makes this algorithm flawed.