Hard 3-CNF-SAT problems are in $P$ -- A first step in proving $NP=P$

TL;DR

This paper proposes a novel lattice framework and a multi-linear descriptor function to solve hard 3-CNF-SAT problems in polynomial time, suggesting a potential breakthrough towards proving $NP=P$.

Contribution

It introduces a new mathematical framework and polynomial algorithm for solving certain 3-CNF-SAT problems, challenging the traditional complexity assumptions.

Findings

Polynomial complexity for computing solution sets of hard 3-CNF-SAT problems.

Development of a greedy polynomial algorithm for solving these problems.

Implication that some NP problems may be in P, impacting the P vs NP question.

Abstract

The relationship between the complexity classes $P$ and $NP$ is an unsolved question in the field of theoretical computer science. In the first part of this paper, a lattice framework is proposed to handle the 3-CNF-SAT problems, known to be in $NP$. In the second section, we define a multi-linear descriptor function ${\cal H}_\varphi$ for any 3-CNF-SAT problem $\varphi$ of size $n$, in the sense that ${\cal H}_\varphi : \{0,1\}^n \rightarrow \{0,1\}^n$ is such that $Im \; {\cal H}_\varphi$ is the set of all the solutions of $\varphi$. A new merge operation ${\cal H}_\varphi \bigwedge {\cal H}_{\psi}$ is defined, where $\psi$ is a single 3-CNF clause. Given ${\cal H}_\varphi$ [but this can be of exponential complexity], the complexity needed for the computation of $Im \; {\cal H}_\varphi$, the set of all solutions, is shown to be polynomial for hard 3-CNF-SAT problems, i.e. the one with few ($\leq 2^k$) or no solutions. The third part uses the relation between ${\cal H}_\varphi$ and the indicator function $\mathbb{1}_{{\cal S}_\varphi}$ for the set of solutions, to develop a greedy polynomial algorithm to solve hard 3-CNF-SAT problems.