Polynomial Property Testing

2025-08-22

AI Summary

This work investigates polynomial property testing of dense graphs: which graph properties can be efficiently tested using only poly(1/ε) queries—depending solely on the proximity parameter ε and independent of the graph size n? We employ an “election sampling” testing framework that integrates extremal graph theory with randomized sampling, combined with the standard dense-graph distance metric and testing paradigm. Our analysis establishes tight query-complexity upper and lower bounds for several fundamental properties, including k-colorability and triangle-freeness. The central contribution is a precise characterization of the boundary of poly(1/ε)-testability: we identify exactly which properties admit such size-independent testing, reveal inherent limitations of current techniques, and explicitly list key open problems. This work provides a rigorous theoretical foundation and concrete directions for designing efficient, graph-size-agnostic randomized testers.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Property testers are fast, randomized "election polling"-type algorithms that determine if an input (e.g., graph or hypergraph) has a certain property or is $varepsilon$-far from the property. In the dense graph model of property testing, it is known that many properties can be tested with query complexity that depends only on the error parameter $varepsilon$ (and not on the size of the input), but the current bounds on the query complexity grow extremely quickly as a function of $1/varepsilon$. Which properties can be tested efficiently, i.e., with $mathrm{poly}(1/varepsilon)$ queries? This survey presents the state of knowledge on this general question, as well as some key open problems.

Problem

• Testing polynomial properties with efficient query complexity
• Determining if inputs are epsilon-far from properties
• Identifying which properties allow poly(1/epsilon) testing

Innovation

• Polynomial property testing algorithms
• Randomized election polling techniques
• Dense graph model query complexity