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Unbounded-width CSPs are Untestable in a Sublinear Number of Queries

Yumou Fei

TL;DR

This work establishes a universal lower bound for testing CSP satisfiability in the bounded-degree model: every unbounded-width CSP template requires $${\Omega}(n)$$ queries, unifying prior lower bounds from distinct CSP cases. The authors build on a universal-algebra framework to show that such templates simulate linear-equation constraints via a structured gadget construction, then reduce hard linear-equation instances (via $3Sum_{\mathsf{G}}$) to bounded-degree CSP instances. The reduction uses an expander/gadget approach to enforce constant relationships and control variable degrees, enabling an unconditional, sublinear-query lower bound that applies to all NP-hard CSPs (e.g., $k$-colorability in $\ell$-uniform hypergraphs for most $(k,\ell)$). The results illuminate the deep connection between sublinear-time testing, universal algebra, and classical hardness, and motivate open problems about tightness, bounded-width testing, and extensions to streaming models.

Abstract

The bounded-degree query model, introduced by Goldreich and Ron (\textit{Algorithmica, 2002}), is a standard framework in graph property testing and sublinear-time algorithms. Many properties studied in this model, such as bipartiteness and 3-colorability of graphs, can be expressed as satisfiability of constraint satisfaction problems (CSPs). We prove that for the entire class of \emph{unbounded-width} CSPs, testing satisfiability requires $Ω(n)$ queries in the bounded-degree model. This result unifies and generalizes several previous lower bounds. In particular, it applies to all CSPs that are known to be $\mathbf{NP}$-hard to solve, including $k$-colorability of $\ell$-uniform hypergraphs for any $k,\ell \ge 2$ with $(k,\ell) \neq (2,2)$. Our proof combines the techniques from Bogdanov, Obata, and Trevisan (\textit{FOCS, 2002}), who established the first $Ω(n)$ query lower bound for CSP testing in the bounded-degree model, with known results from universal algebra.

Unbounded-width CSPs are Untestable in a Sublinear Number of Queries

TL;DR

This work establishes a universal lower bound for testing CSP satisfiability in the bounded-degree model: every unbounded-width CSP template requires queries, unifying prior lower bounds from distinct CSP cases. The authors build on a universal-algebra framework to show that such templates simulate linear-equation constraints via a structured gadget construction, then reduce hard linear-equation instances (via ) to bounded-degree CSP instances. The reduction uses an expander/gadget approach to enforce constant relationships and control variable degrees, enabling an unconditional, sublinear-query lower bound that applies to all NP-hard CSPs (e.g., -colorability in -uniform hypergraphs for most ). The results illuminate the deep connection between sublinear-time testing, universal algebra, and classical hardness, and motivate open problems about tightness, bounded-width testing, and extensions to streaming models.

Abstract

The bounded-degree query model, introduced by Goldreich and Ron (\textit{Algorithmica, 2002}), is a standard framework in graph property testing and sublinear-time algorithms. Many properties studied in this model, such as bipartiteness and 3-colorability of graphs, can be expressed as satisfiability of constraint satisfaction problems (CSPs). We prove that for the entire class of \emph{unbounded-width} CSPs, testing satisfiability requires queries in the bounded-degree model. This result unifies and generalizes several previous lower bounds. In particular, it applies to all CSPs that are known to be -hard to solve, including -colorability of -uniform hypergraphs for any with . Our proof combines the techniques from Bogdanov, Obata, and Trevisan (\textit{FOCS, 2002}), who established the first query lower bound for CSP testing in the bounded-degree model, with known results from universal algebra.

Paper Structure

This paper contains 43 sections, 28 theorems, 66 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Testing CSP Satisfiability
  3. Discussion: Testing vs. Approximation
  4. MaxCSP Problems
  5. MaxCSP vs. Satisfiability Testing
  6. Sublinear Time vs. Polynomial Time
  7. Sublinear Time vs. Sublinear Space
  8. Open Problems
  9. Preliminaries
  10. Relational Structures
  11. Algebraic Structures
  12. Irredundant Relations and Galois Duality
  13. Main Lemma from Universal Algebra
  14. Step 1: construction of $\varphi$.
  15. Step 2: properties of $\varphi$.
  16. ...and 28 more sections

Key Result

Theorem 1.6

For any CSP template $(D,\Gamma)$ of unbounded width, there exist constants $\varepsilon\in (0,1)$ and $d\in \mathbb{N}$ such that any $\varepsilon$-tester for satisfiability of $\mathrm{CSP}(\Gamma)$ instances in the $\mathrm{BD}(d,n)$ model must make $\Omega_{d,\varepsilon}(n)$ queries.

Theorems & Definitions (80)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Definition 1.7
  • Theorem 1.8
  • Theorem 1.9: dalmau2013robust
  • Remark 1.10
  • ...and 70 more