Toward a Uniform Algorithm and Uniform Reduction for Constraint Problems

2026-04-07

AI Summary

This work investigates the expressive power of higher-order algorithms—such as k-consistency, Sherali–Adams linear programming relaxations, and Affine Integer Programming (AIP)—for solving Constraint Satisfaction Problems (CSPs) and their promise variants. By introducing a polymorphic minion structure over templates, the authors establish a unified framework that fully characterizes the capabilities of these algorithms in terms of minion homomorphisms. They further propose a novel hierarchy of orthogonal relaxations based on vectors over ℤₚ, proving its equivalence to AIP-ℤₚ and showing it constitutes a new class of semidefinite programming–like methods. This approach successfully solves the smallest known CSP counterexample resistant to BLP+AIP—the D₄-group CSP—and decides solvability of systems of linear equations modulo p² at prime order p.

Abstract

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 $p^2$.

Problem

• Constraint Satisfaction Problem
• Promise CSP
• Polymorphism Minion
• k-consistency
• Vector Relaxation

Innovation

• constraint satisfaction problem
• Promise CSP
• polymorphism minion
• vector relaxation
• AIP hierarchy