On Triangle Estimation Using Tripartite Independent Set Queries

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 approximate triangle counting algorithm using only polylogarithmic queries when the number of triangles on any edge in the graph is polylogarithmically bounded. Our query oracle Tripartite Independent Set (TIS) takes three disjoint sets of vertices A, B and C as input, 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 group queries (Ron and Tsur, ACM ToCT, 2016; Dell and Lapinskas, STOC 2018) and in particular is inspired by the Bipartite Independent Set (BIS) query oracle of Beame et al. (ITCS 2018). We extend the algorithmic framework of Beame et al., with TIS replacing BIS, for triangle counting using ideas from color coding due to Alon et al. (J. ACM, 1995) and a concentration inequality for sums of random variables with bounded dependency (Janson, Rand. Struct. Alg., 2004). ACM Subject ClassificationTheory of computation → Models of computation; Theory of computation → Streaming, sublinear and near linear time algorithms; Mathematics of computing → Probabilistic algorithms

Approximately counting and sampling small witnesses using a colourful decision oracle

“…When k = 3, our colourful independence oracles are similar to the tripartite independent set (TIS) oracles of Bhattacharya et al [8]. (These oracles ask whether a 3-coloured graph H contains a colourful triangle, rather than whether a 3-coloured 3-hypergraph G contains a colourful edge.…”

Hyperedge Estimation using Polylogarithmic Subset Queries

“…We extend this research direction of parameter estimation problems using subset queries to Hyperedge-Estimation. Our algorithm can be seen as a natural extension of our earlier work on triangle estimation [BBGM18].…”

“…To get around this lower bound, Beame et al [BHR + 18] introduced the BIS query model and estimated the number of edges using polylogarithmic BIS queries. In a farther generalization, triangle estimation with polylogarithmic queries in a graph using a subset query named Tripartite Independent Set query was studied in [BBGM18]. The GPIS subset query that we use in this paper was earlier used to design parameterized query complexities for the hitting set problem [BGK + 18b].…”

Graph Connectivity and Single Element Recovery via Linear and OR Queries

“…Our work falls in the broad class of algorithm design in the query access model, where one has limited access to the input. Over the years there has been a significant amount of work relevant to this paper including in graph reconstruction [1,4,5,7,11,13,14,17,32,45,52], parameter estimation [9,10,21,53], minimum cuts [8,54] sketching and streaming [2,6,8,28,29,37,40,41,47,57], combinatorial group testing, compressed sensing, and coin weighing [11,19,[23][24][25][26]55]. It is impossible to do complete justice, but in Section 7 we discuss some works and how they fit in with our paper.…”

“…The above query models (and similar variants such as additive queries [32], cut-queries [54], edgedetection queries [7,9]) have a rich literature [1,5,9,16,17,32,45,50,54]. Most previous works, however, have focused on either graph reconstruction [14,16,17,45], or on parameter estimation (e.g., estimating the number of edges [9] or triangles [10]). In this work, however, our goal is to understand the power and limitations of these queries to reveal structural properties of the underlying graph.…”