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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2410.19066 |
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| _version_ | 1866909364555612160 |
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| author | Anand, Aditya Lee, Euiwoong Sharma, Amatya |
| author_facet | Anand, Aditya Lee, Euiwoong Sharma, Amatya |
| contents | Given a fixed arity $k \geq 2$, Min-$k$-CSP on complete instances involves a set of $n$ variables $V$ and one nontrivial constraint for every $k$-subset of variables (so there are $\binom{n}{k}$ constraints). The goal is to find an assignment that minimizes unsatisfied constraints. Unlike Max-$k$-CSP that admits a PTAS on dense or expanding instances, the approximability of Min-$k$-CSP is less understood. For some CSPs like Min-$k$-SAT, there's an approximation-preserving reduction from general to dense instances, making complete instances unique for potential new techniques.
This paper initiates a study of Min-$k$-CSPs on complete instances. We present an $O(1)$-approximation algorithm for Min-2-SAT on complete instances, the minimization version of Max-2-SAT. Since $O(1)$-approximation on dense or expanding instances refutes the Unique Games Conjecture, it shows a strict separation between complete and dense/expanding instances.
Then we study the decision versions of CSPs, aiming to satisfy all constraints; which is necessary for any nontrivial approximation. Our second main result is a quasi-polynomial time algorithm for every Boolean $k$-CSP on complete instances, including $k$-SAT. We provide additional algorithmic and hardness results for CSPs with larger alphabets, characterizing (arity, alphabet size) pairs that admit a quasi-polynomial time algorithm on complete instances. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_19066 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Min-CSPs on Complete Instances Anand, Aditya Lee, Euiwoong Sharma, Amatya Data Structures and Algorithms F.2.2; F.2.3; F.2.1 Given a fixed arity $k \geq 2$, Min-$k$-CSP on complete instances involves a set of $n$ variables $V$ and one nontrivial constraint for every $k$-subset of variables (so there are $\binom{n}{k}$ constraints). The goal is to find an assignment that minimizes unsatisfied constraints. Unlike Max-$k$-CSP that admits a PTAS on dense or expanding instances, the approximability of Min-$k$-CSP is less understood. For some CSPs like Min-$k$-SAT, there's an approximation-preserving reduction from general to dense instances, making complete instances unique for potential new techniques. This paper initiates a study of Min-$k$-CSPs on complete instances. We present an $O(1)$-approximation algorithm for Min-2-SAT on complete instances, the minimization version of Max-2-SAT. Since $O(1)$-approximation on dense or expanding instances refutes the Unique Games Conjecture, it shows a strict separation between complete and dense/expanding instances. Then we study the decision versions of CSPs, aiming to satisfy all constraints; which is necessary for any nontrivial approximation. Our second main result is a quasi-polynomial time algorithm for every Boolean $k$-CSP on complete instances, including $k$-SAT. We provide additional algorithmic and hardness results for CSPs with larger alphabets, characterizing (arity, alphabet size) pairs that admit a quasi-polynomial time algorithm on complete instances. |
| title | Min-CSPs on Complete Instances |
| topic | Data Structures and Algorithms F.2.2; F.2.3; F.2.1 |
| url | https://arxiv.org/abs/2410.19066 |