On Triangle Estimation using Tripartite Independent Set Queries

TL;DR

This paper introduces an algorithm for approximately counting triangles in graphs using polylogarithmic queries with a new tripartite independent set query model, extending previous bipartite models.

Contribution

It develops a novel algorithm leveraging tripartite independent set queries to efficiently estimate triangle counts in graphs, especially when local triangle counts are polylogarithmically bounded.

Findings

Achieves polylogarithmic query complexity for triangle estimation.

Extends bipartite query frameworks to tripartite settings.

Provides theoretical bounds for the algorithm's performance.

Abstract

Estimating the number of triangles in a graph is one of the most fundamental problems in sublinear algorithms. In this work, we provide an algorithm that approximately counts the number of triangles in a graph using only polylogarithmic queries when \emph{the number of triangles on any edge in the graph is polylogarithmically bounded}. Our query oracle {\em Tripartite Independent Set} (TIS) takes three disjoint sets of vertices $A$, $B$ and $C$ as inputs, and answers whether there exists a triangle having one endpoint in each of these three sets. Our query model generally belongs to the class of \emph{group queries} (Ron and Tsur, ACM ToCT, 2016; Dell and Lapinskas, STOC 2018) and in particular is inspired by the {\em Bipartite Independent Set} (BIS) query oracle of Beame {\em et al.} (ITCS 2018). We extend the algorithmic framework of Beame {\em et al.}, with \tis replacing \bis, for approximately counting triangles in graphs.