Maximum And- vs. Even-SAT
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866918358649143296 |
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| author | Nakajima, Tamio-Vesa Živný, Stanislav |
| author_facet | Nakajima, Tamio-Vesa Živný, Stanislav |
| contents | A multiset of literals, called a clause, is \emph{strongly satisfied} by an assignment if \emph{no} literal evaluates to false. Finding an assignment that maximises the number of strongly satisfied clauses is NP-hard. We present a simple algorithm that finds, given a multiset of clauses that admits an assignment that strongly satisfies $ρ$ of the clauses, an assignment in which at least $ρ$ of the clauses are \emph{weakly satisfied}, in the sense that an \emph{even} number of literals evaluate to false. In particular, this implies an efficient algorithm for finding an undirected cut of value $ρ$ in a graph $G$ given that a directed cut of value $ρ$ in $G$ is promised to exist. A similar argument also gives an efficient algorithm for finding an acyclic subgraph of $G$ with $ρ$ edges under the same promise. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_07837 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximum And- vs. Even-SAT Nakajima, Tamio-Vesa Živný, Stanislav Data Structures and Algorithms A multiset of literals, called a clause, is \emph{strongly satisfied} by an assignment if \emph{no} literal evaluates to false. Finding an assignment that maximises the number of strongly satisfied clauses is NP-hard. We present a simple algorithm that finds, given a multiset of clauses that admits an assignment that strongly satisfies $ρ$ of the clauses, an assignment in which at least $ρ$ of the clauses are \emph{weakly satisfied}, in the sense that an \emph{even} number of literals evaluate to false. In particular, this implies an efficient algorithm for finding an undirected cut of value $ρ$ in a graph $G$ given that a directed cut of value $ρ$ in $G$ is promised to exist. A similar argument also gives an efficient algorithm for finding an acyclic subgraph of $G$ with $ρ$ edges under the same promise. |
| title | Maximum And- vs. Even-SAT |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2409.07837 |