Hierarchies of Minion Tests for PCSPs through Tensors

We provide a unified framework to study hierarchies of relaxations for Constraint Satisfaction Problems and their Promise variant. The idea is to split the description of a hierarchy into an algebraic part, depending on a minion capturing the “base level” of the hierarchy, and a geometric part - which we call tensorisation - inspired by multilinear algebra. We show that the hierarchies of minion tests obtained in this way are general enough to capture the (combinatorial) bounded width and also the Sherali-Adams LP, Sum-of-Squares SDP, and affine IP hierarchies. We exploit the geometry of the tensor spaces arising from our construction to prove general properties of such hierarchies. We identify certain classes of minions, which we call linear and conic, whose corresponding hierarchies have particularly fine features. Finally, in order to analyse the Sum-of-Squares SDP hierarchy we also characterise the solvability of the standard SDP relaxation through a new minion. * The research leading to these results has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 714532). The paper reflects only the authors' views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein. This work was also supported by UKRI EP/X024431/1. † The full version of the paper can be accessed at https://arxiv.org/abs/2207.02277.