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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.00396 |
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| _version_ | 1866909716017315840 |
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| author | Gaysin, Azza |
| author_facet | Gaysin, Azza |
| contents | Constraint Satisfaction Problems (CSPs) form a broad class of combinatorial problems, which can be formulated as homomorphism problems between relational structures. The CSP dichotomy theorem classifies all such problems over finite domains into two categories: NP-complete and polynomial-time, see Zhuk (2017), Bulatov (2017). Polynomial-time CSPs can be further subdivided into smaller subclasses. Mal'tsev CSPs are defined by the property that every relation in the problem is invariant under a Mal'tsev operation, a ternary operation $μ$ satisfying $μ(x, y, y) = μ(y, y, x) = x$ for all $x, y$. Bulatov and Dalmau proved that Mal'tsev CSPs are solvable in polynomial time, presenting an algorithm for such CSPs (2006). The negation of an unsatisfiable CSP instance can be expressed as a propositional tautology. We formalize the algorithm for Mal'tsev CSPs within bounded arithmetic $V^1$, which captures polynomial-time reasoning and corresponds to the extended Frege proof system. We show that $V^1$ proves the soundness of Mal'tsev algorithm, implying that tautologies expressing the non-existence of a solution for unsatisfiable instances of Mal'tsev CSPs admit short extended Frege proofs. In addition, with small adjustments, we achieved an analogous result for Dalmau's algorithm that solves generalized majority-minority CSPs -- a common generalization of near-unanimity operations and Mal'tsev operations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00396 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Proof complexity of Mal'tsev CSP Gaysin, Azza Logic Computational Complexity Constraint Satisfaction Problems (CSPs) form a broad class of combinatorial problems, which can be formulated as homomorphism problems between relational structures. The CSP dichotomy theorem classifies all such problems over finite domains into two categories: NP-complete and polynomial-time, see Zhuk (2017), Bulatov (2017). Polynomial-time CSPs can be further subdivided into smaller subclasses. Mal'tsev CSPs are defined by the property that every relation in the problem is invariant under a Mal'tsev operation, a ternary operation $μ$ satisfying $μ(x, y, y) = μ(y, y, x) = x$ for all $x, y$. Bulatov and Dalmau proved that Mal'tsev CSPs are solvable in polynomial time, presenting an algorithm for such CSPs (2006). The negation of an unsatisfiable CSP instance can be expressed as a propositional tautology. We formalize the algorithm for Mal'tsev CSPs within bounded arithmetic $V^1$, which captures polynomial-time reasoning and corresponds to the extended Frege proof system. We show that $V^1$ proves the soundness of Mal'tsev algorithm, implying that tautologies expressing the non-existence of a solution for unsatisfiable instances of Mal'tsev CSPs admit short extended Frege proofs. In addition, with small adjustments, we achieved an analogous result for Dalmau's algorithm that solves generalized majority-minority CSPs -- a common generalization of near-unanimity operations and Mal'tsev operations. |
| title | Proof complexity of Mal'tsev CSP |
| topic | Logic Computational Complexity |
| url | https://arxiv.org/abs/2508.00396 |