Polynomial Property Testing

TL;DR

This survey reviews the current understanding of property testing in dense graphs, focusing on which properties can be tested efficiently with polynomial query complexity in terms of 1/ε, and discusses open problems.

Contribution

It provides a comprehensive overview of the state of knowledge and open problems in polynomial property testing, highlighting the gap between known bounds and efficient testing.

Findings

Many properties can be tested with query complexity independent of input size

Current bounds on query complexity grow rapidly with 1/ε

Open problems remain in characterizing efficiently testable properties

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.