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Computational Complexity in Property Testing

Renato Ferreira Pinto, Diptaksho Palit, Sofya Raskhodnikova

TL;DR

The paper investigates the computational hardness of property testing by linking query complexity to time complexity through RAM-based reductions and error-correcting codes. It establishes time–query hierarchies that, unconditionally (weak) or under SETH (strong), separate how many queries testers need from how much time they require, across a spectrum of regimes. A centerpiece is a distribution-free distance approximation problem for halfspaces, for which a fine-grained, $k$-SUM–based reduction yields conditional lower bounds on running time that separate query and time, complemented by unconditional SQ lower bounds under Gaussian inputs. The results illuminate intrinsic gaps between information-theoretic query efficiency and computational effort, and they open avenues for fine-grained hardness results in tolerantly testing and geometric function classes with rigorous complexity barriers.

Abstract

We initiate a systematic study of the computational complexity of property testing, focusing on the relationship between query and time complexity. While traditional work in property testing has emphasized query complexity, relatively little is known about the computational hardness of property testers. Our goal is to chart the landscape of time-query interplay and develop tools for proving time complexity lower bounds. Our first contribution is a pair of time-query hierarchy theorems for property testing. For all suitable nondecreasing functions $q(n)$ and $t(n)$ with $t(n)\geq q(n)$, we construct properties with query complexity $\tildeΘ(q(n))$ and time complexity $\tildeΩ(t(n))$. Our weak hierarchy holds unconditionally, whereas the strong version-assuming the Strong Exponential Time Hypothesis-provides better control over the time complexity of the constructed properties. We then turn to halfspaces in $\mathbb{R}^d$, a fundamental class in property testing and learning theory. We study the problem of approximating the distance from the input function to the nearest halfspace within additive error $ε$. For the distribution-free distance approximation problem, known algorithms achieve query complexity $O(d/ε^2)$, but take time $\tildeΘ(1/ε^d)$. We provide a fine-grained justification for this gap: assuming the $k$-SUM conjecture, any algorithm must have running time $Ω(1/ε^{d/2})$. This fine-grained lower bound yields a provable separation between query and time complexity for a natural and well-studied (tolerant) testing problem. We also prove that any Statistical Query (SQ) algorithm under the standard Gaussian distribution requires $(1/ε)^{Ω(d)}$ queries if the queries are answered with additive error up to $ε^{Ω(d)}$, revealing a fundamental barrier even in the distribution-specific setting.

Computational Complexity in Property Testing

TL;DR

The paper investigates the computational hardness of property testing by linking query complexity to time complexity through RAM-based reductions and error-correcting codes. It establishes time–query hierarchies that, unconditionally (weak) or under SETH (strong), separate how many queries testers need from how much time they require, across a spectrum of regimes. A centerpiece is a distribution-free distance approximation problem for halfspaces, for which a fine-grained, -SUM–based reduction yields conditional lower bounds on running time that separate query and time, complemented by unconditional SQ lower bounds under Gaussian inputs. The results illuminate intrinsic gaps between information-theoretic query efficiency and computational effort, and they open avenues for fine-grained hardness results in tolerantly testing and geometric function classes with rigorous complexity barriers.

Abstract

We initiate a systematic study of the computational complexity of property testing, focusing on the relationship between query and time complexity. While traditional work in property testing has emphasized query complexity, relatively little is known about the computational hardness of property testers. Our goal is to chart the landscape of time-query interplay and develop tools for proving time complexity lower bounds. Our first contribution is a pair of time-query hierarchy theorems for property testing. For all suitable nondecreasing functions and with , we construct properties with query complexity and time complexity . Our weak hierarchy holds unconditionally, whereas the strong version-assuming the Strong Exponential Time Hypothesis-provides better control over the time complexity of the constructed properties. We then turn to halfspaces in , a fundamental class in property testing and learning theory. We study the problem of approximating the distance from the input function to the nearest halfspace within additive error . For the distribution-free distance approximation problem, known algorithms achieve query complexity , but take time . We provide a fine-grained justification for this gap: assuming the -SUM conjecture, any algorithm must have running time . This fine-grained lower bound yields a provable separation between query and time complexity for a natural and well-studied (tolerant) testing problem. We also prove that any Statistical Query (SQ) algorithm under the standard Gaussian distribution requires queries if the queries are answered with additive error up to , revealing a fundamental barrier even in the distribution-specific setting.

Paper Structure

This paper contains 33 sections, 26 theorems, 31 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation: known gaps between query and time complexity
  3. Overview of our results and techniques
  4. Time-query hierarchy.
  5. Prior work on hierarchies in property testing
  6. Open questions
  7. Our model and setup
  8. Computational model
  9. Property testing
  10. Notation
  11. Time-query hierarchies in property testing
  12. Hard languages
  13. Hard language in the RAM model
  14. Hard language from SETH
  15. Efficiently constructable, encodable, and decodable codes
  16. ...and 18 more sections

Key Result

Theorem 3.3

Let $\varepsilon \in (0, 1)$ be a sufficiently small constant. Let $q, t : \mathbb{N} \to \mathbb{N}$ be admissible query and time functions (as in def:admissible-functions) such that $q^{-1}(n)$ is computable in time $O(t \circ q^{-1} (n))$. Then there exists a propertyWe do not try to optimize the

Figures (2)

  • Figure 1: The polygon in the reduction for $d=3$.
  • Figure 2: An illustration of our reduction for $d = 3$ and $n = 2$.

Theorems & Definitions (71)

  • Definition 2.1: Property, distance
  • Definition 2.2: Tester
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1: Admissible query and time functions
  • Definition 3.2: Query and time complexity functions and classes
  • Theorem 3.3: Unconditional weak hierarchy
  • Theorem 3.4: Strong hierarchy from SETH
  • Theorem 3.5: CookR73
  • Theorem 3.6
  • ...and 61 more