Testing Properties of Edge Distributions

TL;DR

This paper explores the problem of testing properties of edge distributions in graphs, providing nearly-tight bounds for key properties like bipartiteness and triangle-freeness, and introduces new techniques based on birthday paradox lemmas.

Contribution

It establishes nearly-tight sample complexity bounds for testing edge distribution properties and develops novel birthday-paradox-based techniques for square-freeness testing.

Findings

Sample complexity for bipartiteness testing is Θ(n)

Triangle-freeness testing requires n^{4/3± o(1)} samples

Square-freeness testing requires n^{9/8± o(1)} samples

Abstract

We initiate the study of distribution testing for probability distributions over the edges of a graph, motivated by the closely related question of ``edge-distribution-free'' graph property testing. The main results of this paper are nearly-tight bounds on testing bipartiteness, triangle-freeness and square-freeness of edge distributions, whose sample complexities are shown to scale as $\Theta(n)$, $n^{4/3\pm o(1)}$ and $n^{9/8\pm o(1)}$, respectively. The technical core of our paper lies in the proof of the upper bound for testing square-freeness, wherein we develop new techniques based on certain birthday-paradox-type lemmas that may be of independent interest. We will discuss how our techniques fit into the general framework of distribution-free property testing. We will also discuss how our results are conceptually connected with Tur\'an problems and subgraph removal lemmas in extremal combinatorics.