A Simple SublinearTime Algorithm for Counting Arbitrary Subgraphs via Edge Sampling
Abstract
In the subgraph counting problem, we are given a input graph $G(V, E)$ and a target graph $H$; the goal is to estimate the number of occurrences of $H$ in $G$. Our focus here is on designing sublineartime algorithms for approximately counting occurrences of $H$ in $G$ in the setting where the algorithm is given query access to $G$. This problem has been studied in several recent papers which primarily focused on specific families of graphs $H$ such as triangles, cliques, and stars. However, not much is known about approximate counting of arbitrary graphs $H$. This is in sharp contrast to the closely related subgraph enumeration problem that has received significant attention in the database community as the database join problem. The AGM bound shows that the maximum number of occurrences of any arbitrary subgraph $H$ in a graph $G$ with $m$ edges is $O(m^{\rho(H)})$, where $\rho(H)$ is the fractional edgecover of $H$, and enumeration algorithms with matching runtime are known for any $H$. We bridge this gap between subgraph counting and subgraph enumeration by designing a sublineartime algorithm that can estimate the number of any arbitrary subgraph $H$ in $G$, denoted by $\#H$, to within a $(1\pm \epsilon)$approximation w.h.p. in $O(\frac{m^{\rho(H)}}{\#H}) \cdot poly(\log{n},1/\epsilon)$ time. Our algorithm is allowed the standard set of queries for general graphs, namely degree queries, pair queries and neighbor queries, plus an additional edgesample query that returns an edge chosen uniformly at random. The performance of our algorithm matches those of Eden et.al. [FOCS 2015, STOC 2018] for counting triangles and cliques and extend them to all choices of subgraph $H$ under the additional assumption of edgesample queries. We further show that our algorithm works for the more general database join size estimation problem and prove a matching lower bound for this problem.
 Publication:

arXiv eprints
 Pub Date:
 November 2018
 arXiv:
 arXiv:1811.07780
 Bibcode:
 2018arXiv181107780A
 Keywords:

 Computer Science  Data Structures and Algorithms