Toward a Uniform Algorithm and Uniform Reduction for Constraint Problems
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914455414112256 |
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| author | Barto, Libor Hadek, Maximilian Zhuk, Dmitriy |
| author_facet | Barto, Libor Hadek, Maximilian Zhuk, Dmitriy |
| contents | We develop a unified framework to characterize the power of higher-level algorithms for the constraint satisfaction problem (CSP), such as $k$-consistency, the Sherali-Adams LP hierarchy, and the affine IP hierarchy. As a result, solvability of a fixed-template CSP or, more generally, a Promise CSP by a given level is shown to depend only on the polymorphism minion of the template. Similarly, we obtain a minion-theoretic description of $k$-consistency reductions between Promise CSPs.
We introduce a new hierarchy of SDP-like vector relaxations with vectors over $\mathbb Z_{p}$ in which orthogonality is imposed on $k$-tuples of vectors. Surprisingly, this relaxation turns out to be equivalent to the $k$-th level of the AIP-$\mathbb{Z}_p$ relaxation. We show that it solves the CSP of the dihedral group $\mathbf{D}_4$, the smallest CSP that fools the singleton BLP+AIP algorithm. Using this vector representation, we further show that the $p$-th level of the $\mathbb{Z}_p$ relaxation solves linear equations modulo $p^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06335 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Toward a Uniform Algorithm and Uniform Reduction for Constraint Problems Barto, Libor Hadek, Maximilian Zhuk, Dmitriy Logic in Computer Science Computational Complexity Data Structures and Algorithms We develop a unified framework to characterize the power of higher-level algorithms for the constraint satisfaction problem (CSP), such as $k$-consistency, the Sherali-Adams LP hierarchy, and the affine IP hierarchy. As a result, solvability of a fixed-template CSP or, more generally, a Promise CSP by a given level is shown to depend only on the polymorphism minion of the template. Similarly, we obtain a minion-theoretic description of $k$-consistency reductions between Promise CSPs. We introduce a new hierarchy of SDP-like vector relaxations with vectors over $\mathbb Z_{p}$ in which orthogonality is imposed on $k$-tuples of vectors. Surprisingly, this relaxation turns out to be equivalent to the $k$-th level of the AIP-$\mathbb{Z}_p$ relaxation. We show that it solves the CSP of the dihedral group $\mathbf{D}_4$, the smallest CSP that fools the singleton BLP+AIP algorithm. Using this vector representation, we further show that the $p$-th level of the $\mathbb{Z}_p$ relaxation solves linear equations modulo $p^2$. |
| title | Toward a Uniform Algorithm and Uniform Reduction for Constraint Problems |
| topic | Logic in Computer Science Computational Complexity Data Structures and Algorithms |
| url | https://arxiv.org/abs/2604.06335 |