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 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 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 group queries (Ron and Tsur ACM Trans. Comput. Theory 8(4), 15, 2016; Dell and Lapinskas 2018) and in particular is inspired by the Bipartite Independent Set (BIS) query oracle of Beame et al. (2018). We extend the algorithmic framework of Beame et al., with TIS replacing BIS, for approximately counting triangles in graphs.
Highlights
Counting the number of triangles in a graph is a fundamental algorithmic problem in the RAM model [4, 10, 22], streaming [1, 2, 5, 11, 13, 24, 25, 26, 27, 28, 33] and the query model [17, 21]
We provide the first approximate triangle counting algorithm using only polylogarithmic queries to a query oracle named Tripartite Independent Set (TIS)
Beame et al [6] precisely did that by estimating the number of edges in a graph using O −4 log14 n bipartite independent set (BIS) queries
Summary
Counting the number of triangles in a graph is a fundamental algorithmic problem in the RAM model [4, 10, 22], streaming [1, 2, 5, 11, 13, 24, 25, 26, 27, 28, 33] and the query model [17, 21]. Bipartite independent set (BIS) queries for a graph, initiated by Beame et al [6], can be seen in the light of group queries It provides a YES/NO answer to the existence of an edge in E(G) that intersects with both V1, V2 ⊂ V (G) of G, where V1 and V2 are disjoint. Beame et al [6] precisely did that by estimating the number of edges in a graph using O −4 log n bipartite independent set (BIS) queries. Motivated by this result, we explore whether triangle estimation can be solved using only polylogarithmic queries to TIS. Both Dell et al [15] and Bhattacharya et al [8], independently, generalized our result to c-uniform hypergraphs, where c ∈ N is a constant
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