Lower Bounds for Approximating Graph Parameters via Communication Complexity

TL;DR

This paper extends a communication complexity technique to derive lower bounds for various graph parameter estimation problems, simplifying proofs and establishing new bounds in different query models.

Contribution

It generalizes existing methods to analyze graph problems across any query type, providing simpler proofs and new lower bounds for several graph parameters.

Findings

New lower bounds for estimating the number of edges and triangles

Simplified proofs for known bounds in graph property testing

Lower bounds that hold in stronger sampling models

Abstract

In a celebrated work, Blais, Brody, and Matulef developed a technique for proving property testing lower bounds via reductions from communication complexity. Their work focused on testing properties of functions, and yielded new lower bounds as well as simplified analyses of known lower bounds. Here, we take a further step in generalizing the methodology of Blais et al. to analyze the query complexity of graph parameter estimation problems. In particular, our technique decouples the lower bound arguments from the representation of the graph, allowing it to work with any query type. We illustrate our technique by providing new simpler proofs of previously known tight lower bounds for the query complexity of several graph problems: estimating the number of edges in a graph, sampling edges from an almost-uniform distribution, estimating the number of triangles (and more generally, $r$-cliques) in a graph, and estimating the moments of the degree distribution of a graph. We also prove new lower bounds for estimating the edge connectivity of a graph and estimating the number of instances of any fixed subgraph in a graph. We show that the lower bounds for estimating the number of triangles and edge connectivity also hold in a strictly stronger computational model that allows access to uniformly random edge samples.