Hard Clique Formulas for Resolution
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908788245659648 |
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| author | Atserias, Albert |
| author_facet | Atserias, Albert |
| contents | We show how to convert any unsatisfiable 3-CNF formula which is sparse and exponentially hard to refute in Resolution into a negative instance of the $k$-clique problem whose corresponding natural encoding as a CNF formula is $n^{Ω(k)}$-hard to refute in Resolution. This applies to any function $k = k(n)$ of the number $n$ of vertices, provided $k_0 \leq k \leq n^{1/c_0}$, where $k_0$ and $c_0$ are small constants. We establish this by demonstrating that Resolution can simulate the correctness proof of a particular kind of reduction from 3-SAT to the parameterized clique problem. This also re-establishes the known conditional hardness result for $k$-clique which states that if the Exponential Time Hypothesis (ETH) holds, then the $k$-clique problem cannot be solved in time $n^{o(k)}$. Since it is known that the analogue of ETH holds for Resolution, unconditionally and with explicit hard instances, this gives a way to obtain explicit instances of $k$-clique that are unconditionally $n^{Ω(k)}$-hard to refute in Resolution. This solves an open problem that appeared published in the literature at least twice. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_12503 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hard Clique Formulas for Resolution Atserias, Albert Computational Complexity Logic in Computer Science We show how to convert any unsatisfiable 3-CNF formula which is sparse and exponentially hard to refute in Resolution into a negative instance of the $k$-clique problem whose corresponding natural encoding as a CNF formula is $n^{Ω(k)}$-hard to refute in Resolution. This applies to any function $k = k(n)$ of the number $n$ of vertices, provided $k_0 \leq k \leq n^{1/c_0}$, where $k_0$ and $c_0$ are small constants. We establish this by demonstrating that Resolution can simulate the correctness proof of a particular kind of reduction from 3-SAT to the parameterized clique problem. This also re-establishes the known conditional hardness result for $k$-clique which states that if the Exponential Time Hypothesis (ETH) holds, then the $k$-clique problem cannot be solved in time $n^{o(k)}$. Since it is known that the analogue of ETH holds for Resolution, unconditionally and with explicit hard instances, this gives a way to obtain explicit instances of $k$-clique that are unconditionally $n^{Ω(k)}$-hard to refute in Resolution. This solves an open problem that appeared published in the literature at least twice. |
| title | Hard Clique Formulas for Resolution |
| topic | Computational Complexity Logic in Computer Science |
| url | https://arxiv.org/abs/2601.12503 |