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  "path": "/abs/2604.06335v1",
  "publishedAt": "2026-04-09T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Libor Barto",
    "Maximilian Hadek",
    "Dmitriy Zhuk"
  ],
  "textContent": "**Authors:** Libor Barto, Maximilian Hadek, Dmitriy Zhuk\n\nWe 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"
}