🤖 AI Summary
This work investigates the expressive power limits of polynomial-time algorithms based on affine integer relaxations—specifically Z-affine k-consistency and the combined BLP+AIP method—for Constraint Satisfaction Problems (CSPs) over finite domains. Constructing the first explicit counterexample, we refute the conjecture that BLP+AIP solves all tractable CSPs and disprove the Dalmau–Opršal dichotomy conjecture equating Z-affine consistency with Datalog reducibility. Theoretical analysis shows that such affine-relaxation methods cannot capture all Maltsev CSPs and thus do not constitute a universal polynomial-time CSP solver. Although co-clone k-consistency resolves our counterexample, it fails on some NP-complete templates. Our results expose fundamental limitations of affine relaxation techniques and clarify critical boundaries among algebraic structure, logical reducibility, and convex optimization relaxations in CSP complexity theory.
📝 Abstract
We show that various recent algorithms for finite-domain constraint satisfaction problems (CSP), which are based on solving their affine integer relaxations, do not solve all tractable and not even all Maltsev CSPs. This rules them out as candidates for a universal polynomial-time CSP algorithm. The algorithms are $mathbb{Z}$-affine $k$-consistency, BLP+AIP, BA$^{k}$, and CLAP. We thereby answer a question by Brakensiek, Guruswami, Wrochna, and v{Z}ivn'y whether BLP+AIP solves all tractable CSPs in the negative. We also refute a conjecture by Dalmau and Oprv{s}al (LICS 2024) that every CSP is either solved by $mathbb{Z}$-affine $k$-consistency or admits a Datalog reduction from 3-colorability. For the cohomological $k$-consistency algorithm, that is also based on affine relaxations, we show that it correctly solves our counterexample but fails on an NP-complete template.