An Efficient Algorithm for Solving the 2-MAXSAT Problem
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916876175540224 |
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| author | Chen, Yangjun |
| author_facet | Chen, Yangjun |
| contents | By the MAXSAT problem, we are given a set $V$ of $m$ variables and a collection $C$ of $n$ clauses over $V$, i.e., a conjunctive normal form ($\textit{CNF}$) formula. We will seek a truth assignment to maximize the number of satisfied clauses in $C$. This problem is $\textit{NP}$-complete even for its restricted version, the 2-maxsat problem, by which every clause contains at most 2 literals. In this paper, we discuss an efficient algorithm to solve this problem. Its main idea is to transform the 2-maxsat problem into a related problem of maximizing satisfied conjunctions in a disjunctive normal form ($\textit{DNF}$) formula $D$. We then represent all those truth assignments for a conjunction $d$ as a graph (called a $p$*-graph), under each of which $d$ evaluates to $\textit{true}$, and organize all the $p$*-graphs for the conjunctions in $D$ into a trie-like structure. By exploring the structure and recursively its substructures (with each corresponding to a subgraph dynamically built up by integrating some $p$*-subgraphs), the algorithm can find a maximum set of satisfied conjunctions in $D$ in polynomial time. Its worst-case time complexity is bounded by O($n^2m^4$). This provides in fact a proof of $P$ = $\textit{NP}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_12517 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An Efficient Algorithm for Solving the 2-MAXSAT Problem Chen, Yangjun Computational Complexity By the MAXSAT problem, we are given a set $V$ of $m$ variables and a collection $C$ of $n$ clauses over $V$, i.e., a conjunctive normal form ($\textit{CNF}$) formula. We will seek a truth assignment to maximize the number of satisfied clauses in $C$. This problem is $\textit{NP}$-complete even for its restricted version, the 2-maxsat problem, by which every clause contains at most 2 literals. In this paper, we discuss an efficient algorithm to solve this problem. Its main idea is to transform the 2-maxsat problem into a related problem of maximizing satisfied conjunctions in a disjunctive normal form ($\textit{DNF}$) formula $D$. We then represent all those truth assignments for a conjunction $d$ as a graph (called a $p$*-graph), under each of which $d$ evaluates to $\textit{true}$, and organize all the $p$*-graphs for the conjunctions in $D$ into a trie-like structure. By exploring the structure and recursively its substructures (with each corresponding to a subgraph dynamically built up by integrating some $p$*-subgraphs), the algorithm can find a maximum set of satisfied conjunctions in $D$ in polynomial time. Its worst-case time complexity is bounded by O($n^2m^4$). This provides in fact a proof of $P$ = $\textit{NP}$. |
| title | An Efficient Algorithm for Solving the 2-MAXSAT Problem |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2304.12517 |