Abstract
We present a simple sublineartime algorithm for sampling an arbitrary subgraph H exactly uniformly from a graph G, to which the algorithm has access by performing the following types of queries: (1) uniform vertex queries, (2) degree queries, (3) neighbor queries, (4) pair queries and (5) edge sampling queries. The query complexity and running time of our algorithm are Õ(min{m, (m^ρ(H))/#H}) and Õ((m^ρ(H))/#H}), respectively, where ρ(H) is the fractional edgecover of H and #H is the number of copies of H in G. For any clique on r vertices, i.e., H = K_r, our algorithm is almost optimal as any algorithm that samples an H from any distribution that has Ω(1) total probability mass on the set of all copies of H must perform Ω(min{m, (m^ρ(H))/(#H⋅(cr)^r)}) queries.
Together with the query and time complexities of the (1±ε)approximation algorithm for the number of subgraphs H by Assadi et al. [Sepehr Assadi et al., 2018] and the lower bound by Eden and Rosenbaum [Eden and Rosenbaum, 2018] for approximately counting cliques, our results suggest that in our query model, approximately counting cliques is "equivalent to" exactly uniformly sampling cliques, in the sense that the query and time complexities of exactly uniform sampling and randomized approximate counting are within polylogarithmic factor of each other. This stands in interesting contrast to an analogous relation between approximate counting and almost uniformly sampling for selfreducible problems in the polynomialtime regime by Jerrum, Valiant and Vazirani [Jerrum et al., 1986].
BibTeX  Entry
@InProceedings{fichtenberger_et_al:LIPIcs:2020:12452,
author = {Hendrik Fichtenberger and Mingze Gao and Pan Peng},
title = {{Sampling Arbitrary Subgraphs Exactly Uniformly in Sublinear Time}},
booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
pages = {45:145:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771382},
ISSN = {18688969},
year = {2020},
volume = {168},
editor = {Artur Czumaj and Anuj Dawar and Emanuela Merelli},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12452},
URN = {urn:nbn:de:0030drops124526},
doi = {10.4230/LIPIcs.ICALP.2020.45},
annote = {Keywords: Graph sampling, Graph algorithms, Sublineartime algorithms}
}
Keywords: 

Graph sampling, Graph algorithms, Sublineartime algorithms 
Collection: 

47th International Colloquium on Automata, Languages, and Programming (ICALP 2020) 
Issue Date: 

2020 
Date of publication: 

29.06.2020 