Graph Property Testing and the Container Method

Abstract

Graph property testing algorithms aim to distinguish between graphs that have a specific property and graphs that are far from the property by inspecting a small random portion of the graph. This thesis studies graph property testing in the dense graph model, where graphs are represented by their adjacency matrices, and a graph is $\epsilon$-far from a property if an $\epsilon$-fraction of the entries of its adjacency matrix must be modified to make the graph have the property. A central goal in the dense graph model is to determine the minimum size subgraph that must be sampled to test natural graph properties.

This thesis develops a new framework for analyzing graph property testing algorithms using the graph and hypergraph container method. Although the container method has become a powerful tool throughout extremal combinatorics, prior to this thesis it had not been used to analyze property testing algorithms. We establish a connection between graph property testing and the container method by showing that suitable container lemmas imply strong upper bounds on the sample complexity of canonical property testers for a number of natural properties.

To demonstrate the framework, we develop new graph and hypergraph container lemmas and apply them to three classic property testing problems. First, we prove a new graph container lemma for graphs that are far from having a $\rho n$ independent set, and use it to give the optimal, up to logarithmic factors, upper bound on the sample complexity of testing the property of having a $\rho n$ independent set. Second, we prove a new hypergraph container lemma for constraint satisfaction problems. This yields a new upper bound on the sample complexity for testing satisfiability, and is the first upper bound that is polynomial in the constraint arity, alphabet size, and distance parameter simultaneously. Finally, we introduce a new container lemma for sparse subgraphs and apply it to tolerant testing the $\rho n$ independent set property, giving an optimal sample complexity bound, again up to logarithmic factors, and demonstrating that container methods can be extended beyond independent sets to sparse subgraphs.

Together, these results demonstrate that the container method provides a general framework for analyzing efficient property testing algorithms.