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 Živný whether BLP+AIP solves all tractable CSPs in the negative. We also refute a conjecture by Dalmau and Opršal (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.

翻译：我们证明，近期提出的多种基于仿射整数松弛求解有限域约束满足问题（CSP）的算法，并不能解决所有可处理的CSP，甚至无法解决所有Maltsev CSP。这排除了它们作为通用多项式时间CSP算法的可能性。这些算法包括$\mathbb{Z}$-仿射$k$-一致性、BLP+AIP、BA$^{k}$和CLAP。由此，我们对Brakensiek、Guruswami、Wrochna和Živný提出的“BLP+AIP是否能够解决所有可处理CSP”这一问题给出了否定回答。同时，我们反驳了Dalmau和Opršal（LICS 2024）的猜想，即每个CSP要么可通过$\mathbb{Z}$-仿射$k$-一致性求解，要么允许从3-可着色性问题进行Datalog归约。对于同样基于仿射松弛的上同调$k$-一致性算法，我们证明其能正确解决我们的反例，但在一个NP完全模板上失效。