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On the complexity of symmetric vs. functional PCSPs

Tamio-Vesa Nakajima, Stanislav Živný

TL;DR

This work advances the understanding of the complexity of promise constraint satisfaction problems PCSP$(\mathbf{A},\mathbf{B})$ by focusing on symmetric $\mathbf{A}$ and functional $\mathbf{B}$. It introduces the notions of additivity and dependency, develops a minion-based, sandwich-style framework, and proves a dichotomy: when $(\mathbf{A},\mathbf{B})$ is additive and dependent, the problem is either solvable in polynomial time by the $AIP$ relaxation (and finitely tractable) or NP-hard. The paper then derives corollaries for Boolean templates and small-arity cases, and proves a collapse result showing $BLP+AIP$ does not exceed the power of $AIP$ for templates with a single balanced relation. Collectively, these results clarify tractable versus hard fragments of PCSPs and provide concrete algorithmic and hardness criteria, with implications for problems such as approximate coloring and linear-algebraic relaxation methods. The methodology combines algebraic minion theory, relaxations, and constructive sandwich arguments to yield precise dichotomy statements and a roadmap for extending tractability classifications to broader PCSP classes.

Abstract

The complexity of the promise constraint satisfaction problem $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$ is largely unknown, even for symmetric $\mathbf{A}$ and $\mathbf{B}$, except for the case when $\mathbf{A}$ and $\mathbf{B}$ are Boolean. First, we establish a dichotomy for $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$ where $\mathbf{A}, \mathbf{B}$ are symmetric, $\mathbf{B}$ is functional (i.e. any $r-1$ elements of an $r$-ary tuple uniquely determines the last one), and $(\mathbf{A},\mathbf{B})$ satisfies technical conditions we introduce called dependency and additivity. This result implies a dichotomy for $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$ with $\mathbf{A},\mathbf{B}$ symmetric and $\mathbf{B}$ functional if (i) $\mathbf{A}$ is Boolean, or (ii) $\mathbf{A}$ is a hypergraph of a small uniformity, or (iii) $\mathbf{A}$ has a relation $R^{\mathbf{A}}$ of arity at least 3 such that the hypergraph diameter of $(A, R^{\mathbf{A}})$ is at most 1. Second, we show that for $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$, where $\mathbf{A}$ and $\mathbf{B}$ contain a single relation, $\mathbf{A}$ satisfies a technical condition called balancedness, and $\mathbf{B}$ is arbitrary, the combined basic linear programming relaxation (BLP) and the affine integer programming relaxation (AIP) is no more powerful than the (in general strictly weaker) AIP relaxation. Balanced $\mathbf{A}$ include symmetric $\mathbf{A}$ or, more generally, $\mathbf{A}$ preserved by a transitive permutation group.

On the complexity of symmetric vs. functional PCSPs

TL;DR

This work advances the understanding of the complexity of promise constraint satisfaction problems PCSP by focusing on symmetric and functional . It introduces the notions of additivity and dependency, develops a minion-based, sandwich-style framework, and proves a dichotomy: when is additive and dependent, the problem is either solvable in polynomial time by the relaxation (and finitely tractable) or NP-hard. The paper then derives corollaries for Boolean templates and small-arity cases, and proves a collapse result showing does not exceed the power of for templates with a single balanced relation. Collectively, these results clarify tractable versus hard fragments of PCSPs and provide concrete algorithmic and hardness criteria, with implications for problems such as approximate coloring and linear-algebraic relaxation methods. The methodology combines algebraic minion theory, relaxations, and constructive sandwich arguments to yield precise dichotomy statements and a roadmap for extending tractability classifications to broader PCSP classes.

Abstract

The complexity of the promise constraint satisfaction problem is largely unknown, even for symmetric and , except for the case when and are Boolean. First, we establish a dichotomy for where are symmetric, is functional (i.e. any elements of an -ary tuple uniquely determines the last one), and satisfies technical conditions we introduce called dependency and additivity. This result implies a dichotomy for with symmetric and functional if (i) is Boolean, or (ii) is a hypergraph of a small uniformity, or (iii) has a relation of arity at least 3 such that the hypergraph diameter of is at most 1. Second, we show that for , where and contain a single relation, satisfies a technical condition called balancedness, and is arbitrary, the combined basic linear programming relaxation (BLP) and the affine integer programming relaxation (AIP) is no more powerful than the (in general strictly weaker) AIP relaxation. Balanced include symmetric or, more generally, preserved by a transitive permutation group.
Paper Structure (18 sections, 49 theorems, 51 equations)

This paper contains 18 sections, 49 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Preliminaries
  4. Structures and PCSPs
  5. Operations and polymorphisms
  6. Relaxations
  7. Minions
  8. Additivity and dependency
  9. Additivity
  10. Dependency
  11. Formal system
  12. Super-connectedness
  13. Dichotomy
  14. Proof of Theorem 42
  15. Proof of Theorem 43
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\mathbf{A}$ be a symmetric structure and $\mathbf{B}$ be a functional structure such that $\mathbf{A} \to \mathbf{B}$. Assume that $(\mathbf{A}, \mathbf{B})$ is dependent and additive. Then, either $\mathop{\mathrm{PCSP}}\nolimits(\mathbf{A}, \mathbf{B})$ is solvable in polynomial time by $\ope

Theorems & Definitions (106)

  • Theorem 1
  • Corollary 1
  • Corollary 1
  • Corollary 1
  • Conjecture 2
  • Theorem 3
  • Corollary 3
  • Theorem 4: BBKO21
  • Theorem 5: BGWZ
  • Lemma 6
  • ...and 96 more