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Hierarchies of Minion Tests for PCSPs through Tensors

Lorenzo Ciardo, Stanislav Živný

TL;DR

In order to analyse the Sum-of-Squares SDP hierarchy, the solvability of the standard SDP relaxation is characterised through a new minion, and the geometry of the tensor spaces arising from the construction are exploited to prove general properties of hierarchies.

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", and a geometric part - which we call tensorisation - inspired by multilinear algebra. We exploit the geometry of the tensor spaces arising from our construction to prove general properties of hierarchies. We identify certain classes of minions, which we call linear and conic, whose corresponding hierarchies have particularly fine features. We establish that the (combinatorial) bounded width, Sherali-Adams LP, affine IP, Sum-of-Squares SDP, and combined "LP + affine IP" hierarchies are all captured by this framework. In particular, in order to analyse the Sum-of-Squares SDP hierarchy, we also characterise the solvability of the standard SDP relaxation through a new minion.

Hierarchies of Minion Tests for PCSPs through Tensors

TL;DR

In order to analyse the Sum-of-Squares SDP hierarchy, the solvability of the standard SDP relaxation is characterised through a new minion, and the geometry of the tensor spaces arising from the construction are exploited to prove general properties of hierarchies.

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", and a geometric part - which we call tensorisation - inspired by multilinear algebra. We exploit the geometry of the tensor spaces arising from our construction to prove general properties of hierarchies. We identify certain classes of minions, which we call linear and conic, whose corresponding hierarchies have particularly fine features. We establish that the (combinatorial) bounded width, Sherali-Adams LP, affine IP, Sum-of-Squares SDP, and combined "LP + affine IP" hierarchies are all captured by this framework. In particular, in order to analyse the Sum-of-Squares SDP hierarchy, we also characterise the solvability of the standard SDP relaxation through a new minion.
Paper Structure (29 sections, 36 theorems, 95 equations, 1 figure)

This paper contains 29 sections, 36 theorems, 95 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contributions
  3. Subsequent work
  4. Organisation
  5. Preliminaries
  6. Notation
  7. Terminology for tensors
  8. Tuples
  9. Semirings
  10. Tensors
  11. Contraction
  12. Promise CSPs
  13. Structures
  14. Polymorphisms and minions
  15. Relaxations and hierarchies
  16. ...and 14 more sections

Key Result

Theorem 6

$\mathop{\mathrm{AC}}\nolimits= \operatorname{Test}_{\mathscr{H}}$, $\mathop{\mathrm{BLP}}\nolimits= \operatorname{Test}_{{\mathscr{Q}_{\operatorname{conv}}}}$, $\mathop{\mathrm{AIP}}\nolimits= \operatorname{Test}_{{\mathscr{Z}_{\operatorname{aff}}}}$, $\mathop{\mathrm{BA}}\nolimits= \operatorname{T

Figures (1)

  • Figure 1: A tensor $M\in R^{\textbf{F}}$ from Example \ref{['example_free_structure']}, corresponding to the uniform distribution on the set of edges of $\mathbf{K}_3$. The opacity of a cell is proportional to the value of the corresponding entry: = $\frac{1}{3}$, = $\frac{1}{6}$, = $0$.

Theorems & Definitions (91)

  • Example 1
  • Example 2
  • Example 3
  • Remark 4
  • Definition 5
  • Theorem 6: BBKO21,bgwz20
  • Proposition 7
  • Lemma 8: BBKO21
  • proof
  • Proposition 9
  • ...and 81 more