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Algebraic Global Gadgetry for Surjective Constraint Satisfaction

Hubie Chen

TL;DR

This work proposes an algebraic framework for surjective CSPs that constructs global gadgetry to encode all assignments $V\to D$ via instances $\Phi_V$, enabling reductions from classical CSPs to surjective CSPs $\mathsf{SCSP}(\mathbf{B})$. It systematically derives NP-hardness results for key problems, including $\mathsf{SCSP}(\mathbf{C})$ (the reflexive $4$-cycle) and $\mathsf{SCSP}(\mathbf{N})$ (no-rainbow $3$-coloring), and, under diagonal-cautious polymorphisms, hardness for all intractable $2$-element target structures; it further provides decidability criteria for stability and explores condensations and sparsifiability. By unifying disparate hardness proofs under a single framework, the paper clarifies how classical CSPs reduce to surjective variants and exposes the common structural principles underlying these reductions. The results offer new sparsification bounds and deepen the algebraic understanding of global gadgetry in surjective CSPs, with implications for future hardness proofs and problem reductions.

Abstract

The constraint satisfaction problem (CSP) on a finite relational structure B is to decide, given a set of constraints on variables where the relations come from B, whether or not there is a assignment to the variables satisfying all of the constraints; the surjective CSP is the variant where one decides the existence of a surjective satisfying assignment onto the universe of B. We present an algebraic framework for proving hardness results on surjective CSPs; essentially, this framework computes global gadgetry that permits one to present a reduction from a classical CSP to a surjective CSP. We show how to derive a number of hardness results for surjective CSP in this framework, including the hardness of the disconnected cut problem, of the no-rainbow 3-coloring problem, and of the surjective CSP on all 2-element structures known to be intractable (in this setting). Our framework thus allows us to unify these hardness results, and reveal common structure among them; we believe that our hardness proof for the disconnected cut problem is more succinct than the original. In our view, the framework also makes very transparent a way in which classical CSPs can be reduced to surjective CSPs.

Algebraic Global Gadgetry for Surjective Constraint Satisfaction

TL;DR

This work proposes an algebraic framework for surjective CSPs that constructs global gadgetry to encode all assignments via instances , enabling reductions from classical CSPs to surjective CSPs . It systematically derives NP-hardness results for key problems, including (the reflexive -cycle) and (no-rainbow -coloring), and, under diagonal-cautious polymorphisms, hardness for all intractable -element target structures; it further provides decidability criteria for stability and explores condensations and sparsifiability. By unifying disparate hardness proofs under a single framework, the paper clarifies how classical CSPs reduce to surjective variants and exposes the common structural principles underlying these reductions. The results offer new sparsification bounds and deepen the algebraic understanding of global gadgetry in surjective CSPs, with implications for future hardness proofs and problem reductions.

Abstract

The constraint satisfaction problem (CSP) on a finite relational structure B is to decide, given a set of constraints on variables where the relations come from B, whether or not there is a assignment to the variables satisfying all of the constraints; the surjective CSP is the variant where one decides the existence of a surjective satisfying assignment onto the universe of B. We present an algebraic framework for proving hardness results on surjective CSPs; essentially, this framework computes global gadgetry that permits one to present a reduction from a classical CSP to a surjective CSP. We show how to derive a number of hardness results for surjective CSP in this framework, including the hardness of the disconnected cut problem, of the no-rainbow 3-coloring problem, and of the surjective CSP on all 2-element structures known to be intractable (in this setting). Our framework thus allows us to unify these hardness results, and reveal common structure among them; we believe that our hardness proof for the disconnected cut problem is more succinct than the original. In our view, the framework also makes very transparent a way in which classical CSPs can be reduced to surjective CSPs.

Paper Structure

This paper contains 15 sections, 18 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Contributions
  4. Preliminaries
  5. Structures, formulas, and problems
  6. Definability and algebra
  7. Framework
  8. Hardness results
  9. Disconnected cut: the reflexive $4$-cycle
  10. No-rainbow $3$-coloring
  11. Diagonal-cautious clones
  12. Establishing stability
  13. Condensations
  14. Sparsifiability
  15. Acknowledgements.

Key Result

Theorem 2.1

Romov81-algebras-partial-functions Let $\mathbf{B}$ be a structure. Let $T$ be a set of non-empty mappings from a finite set $I$ to $B$; for each $i \in I$, let $\pi_i: T \to B$ denote the map defined by $\pi_i(t) = t(i)$. The set $\langle T \rangle_{\mathbf{B}}$ is equal to the set $T'$ of maps fro

Theorems & Definitions (22)

  • Theorem 2.1
  • Proposition 3.1
  • Example 3.2
  • Proposition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Theorem 3.7
  • Theorem 3.8
  • Theorem 4.1
  • ...and 12 more