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Lower Bounds for Approximating Graph Parameters via Communication Complexity

Authors Talya Eden, Will Rosenbaum

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Author Details

Talya Eden
  • School of Electrical Engineering, Tel Aviv University, Tel Aviv 6997801, Israel
Will Rosenbaum
  • School of Electrical Engineering, Tel Aviv University, Tel Aviv 6997801, Israel

Funding

  • Eden, Talya: This research was partially supported by a grant from the Blavatnik fund. The author is grateful to the Azrieli Foundation for the award of an Azrieli Fellowship, and for the Weinstein Institute for their support.

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Talya Eden and Will Rosenbaum. Lower Bounds for Approximating Graph Parameters via Communication Complexity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.11

Abstract

In a celebrated work, Blais, Brody, and Matulef [Blais et al., 2012] 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., 2012] 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Lower bounds and information complexity
Keywords
  • sublinear graph parameter estimation
  • lower bounds
  • communication complexity

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