Abstract
In the subgraph counting problem, we are given a (large) input graph G(V, E) and a (small) target graph H (e.g., a triangle); the goal is to estimate the number of occurrences of H in G. Our focus here is on designing sublineartime algorithms for approximately computing number of 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 in the literature. 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 simple sublineartime algorithm that can estimate the number of occurrences of any arbitrary graph H in G, denoted by #H, to within a (1 +/ epsilon)approximation with high probability in O(m^{rho(H)}/#H) * 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.
BibTeX  Entry
@InProceedings{assadi_et_al:LIPIcs:2018:10099,
author = {Sepehr Assadi and Michael Kapralov and Sanjeev Khanna},
title = {{A Simple SublinearTime Algorithm for Counting Arbitrary Subgraphs via Edge Sampling}},
booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
pages = {6:16:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770958},
ISSN = {18688969},
year = {2018},
volume = {124},
editor = {Avrim Blum},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/10099},
URN = {urn:nbn:de:0030drops100996},
doi = {10.4230/LIPIcs.ITCS.2019.6},
annote = {Keywords: Sublineartime algorithms, Subgraph counting, AGM bound}
}
Keywords: 

Sublineartime algorithms, Subgraph counting, AGM bound 
Seminar: 

10th Innovations in Theoretical Computer Science Conference (ITCS 2019) 
Issue Date: 

2018 
Date of publication: 

21.12.2018 