Limitations of Affine Integer Relaxations for Solving Constraint Satisfaction Problems

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.