Hierarchies of Minion Tests for PCSPs through Tensors

Abstract

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.

Approximate Graph Colouring and Crystals

“…The tensor power used here is unrelated to that notion -in particular, as it is clear from Definition 9, the k-th tensor power of a digraph is not a digraph for k > 1. 6 In particular, the number of hyperedges in H k ○ is equal to the number of edges in H. 7 The result in [29] is proved for arbitrary relational structures; for the purpose of this work, the less general version concerning digraphs is enough. Moreover, the definition of the AIP hierarchy and the other hierarchies characterised in [29] is formally different from the definition used here, in that it requires preprocessing the PCSP template and instance by "k-enhancing" them, i.e., adding dummy constraints on k-tuples of variables.…”

“…6 In particular, the number of hyperedges in H k ○ is equal to the number of edges in H. 7 The result in [29] is proved for arbitrary relational structures; for the purpose of this work, the less general version concerning digraphs is enough. Moreover, the definition of the AIP hierarchy and the other hierarchies characterised in [29] is formally different from the definition used here, in that it requires preprocessing the PCSP template and instance by "k-enhancing" them, i.e., adding dummy constraints on k-tuples of variables. As proved in [29, Section A.1], that definition is equivalent to the more standard hierarchy definition used in [25], which we follow in this work.…”

“…In [29], the AIP hierarchy was characterised algebraically by using a multilinear construction. We state the characterisation in Theorem 11 below, after introducing the necessary terminology.…”

“…Contributions In this paper, we focus on the second class of algorithms and show that no level of the affine integer programming hierarchy solves AGC. Recently, [29] described a linear-algebraic characterisation of the algorithm in terms of a geometric construction called tensorisation. Using this characterisation as a black box, we translate the problem of finding an instance of AGC fooling the algorithm into the problem of finding a tensor with many symmetries, which we call a crystal.…”

Local consistency as a reduction between constraint satisfaction problems

“…Second, Ciardo and Živný [CŽ23] de ned a general framework for hierarchies of algorithms for promise CSPs generalising the Sherali-Adams hierarchy. Again, their hierarchies fall within our framework in the sense that a problem is solved by some level of the Ciardo-Živný hierarchy if and only if it reduces to a corresponding (promise) CSP by a consistency reduction.…”