A simple sublinear-time algorithm for counting arbitrary subgraphs via edge sampling

Sepehr Assadi, Michael Kapralov, Sanjeev Khanna

Research output: Chapter in Book/Report/Conference proceedingConference contribution

14 Scopus citations

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 sublinear-time 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(mρ(H)), where ρ(H) is the fractional edge-cover 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 sublinear-time algorithm that can estimate the number of occurrences of any arbitrary graph H in G, denoted by #H, to within a (1 ± ε)-approximation with high probability in O(m#ρ(HH) ) · poly(log n, 1/ε) time. Our algorithm is allowed the standard set of queries for general graphs, namely degree queries, pair queries and neighbor queries, plus an additional edge-sample 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 edge-sample queries.

Original languageEnglish (US)
Title of host publication10th Innovations in Theoretical Computer Science, ITCS 2019
EditorsAvrim Blum
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770958
DOIs
StatePublished - Jan 1 2019
Externally publishedYes
Event10th Innovations in Theoretical Computer Science, ITCS 2019 - San Diego, United States
Duration: Jan 10 2019Jan 12 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume124
ISSN (Print)1868-8969

Conference

Conference10th Innovations in Theoretical Computer Science, ITCS 2019
Country/TerritoryUnited States
CitySan Diego
Period1/10/191/12/19

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • AGM bound
  • Subgraph counting
  • Sublinear-time algorithms

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