We show that various known algorithms for finite-domain constraint satisfaction problems (CSP), which are based on solving systems of linear equations over the integers, fail to solve all tractable CSPs correctly. The algorithms include $\mathbb{Z}$-affine $k$-consistency, BLP+AIP, every fixed level of the BA$^{k}$-hierarchy, and the CLAP algorithm. In particular, we refute the conjecture by Dalmau and Opr\v{s}al that there is a fixed constant $k$ such that the $\mathbb{Z}$-affine $k$-consistency algorithm solves all tractable finite domain CSPs.

翻译：我们证明，基于求解整数线性方程组的各种已知有限域约束满足问题（CSP）算法，均无法正确求解所有可处理的CSP。这些算法包括$\mathbb{Z}$-仿射$k$-一致性、BLP+AIP、BA$^{k}$-层次结构的任意固定层级以及CLAP算法。特别地，我们否定了Dalmau与Opr\v{s}al的猜想——即存在某个固定常数$k$，使得$\mathbb{Z}$-仿射$k$-一致性算法能够求解所有可处理的有限域CSP。