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Hard 3-CNF-SAT problems are in $P$ -- A first step in proving $NP=P$
arXiv - CS - Computational Complexity Pub Date : 2020-01-03 , DOI: arxiv-2001.00760
Marcel R\'emon and Johan Barth\'elemy

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.

中文翻译:

困难的 3-CNF-SAT 问题在 $P$ 中——证明 $NP=P$ 的第一步

复杂度类$P$和$NP$之间的关系是理论计算机科学领域的一个未解决的问题。在本文的第一部分,提出了一个格框架来处理 3-CNF-SAT 问题,已知在 $NP$ 中。在第二部分中,我们为任何大小为 $n$ 的 3-CNF-SAT 问题 $\varphi$ 定义了一个多线性描述符函数 ${\cal H}_\varphi$,因为 ${\cal H }_\varphi : \{0,1\}^n \rightarrow \{0,1\}^n$ 是这样的 $Im \; {\cal H}_\varphi$ 是 $\varphi$ 的所有解的集合。定义了一个新的合并操作 ${\cal H}_\varphi \bigwedge {\cal H}_{\psi}$,其中 $\psi$ 是单个 3-CNF 子句。给定 ${\cal H}_\varphi$ [但这可能具有指数复杂性],计算 $Im \ 所需的复杂性;{\cal H}_\varphi$,所有解的集合,对于困难的 3-CNF-SAT 问题显示为多项式,即具有很少 ($\leq 2^k$) 或没有解决方案的问题。第三部分使用 ${\cal H}_\varphi$ 与指标函数 $\mathbb{1}_{{\cal S}_\varphi}$ 的关系为解集,推导出贪婪多项式算法来解决困难的 3-CNF-SAT 问题。
更新日期:2020-01-06
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