文章基本信息

标题：Sampling Arbitrary Subgraphs Exactly Uniformly in Sublinear Time
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作者：Hendrik Fichtenberger ; Mingze Gao ; Pan Peng 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：168
页码：45:1-45:13
DOI：10.4230/LIPIcs.ICALP.2020.45
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：We present a simple sublinear-time 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 OÌf(min{m, (m^Ï(H))/#H}) and OÌf((m^Ï(H))/#H}), respectively, where Ï(H) is the fractional edge-cover 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 Î©(1) total probability mass on the set of all copies of H must perform Î©(min{m, (m^Ï(H))/(#Hâ<.(cr)^r)}) queries. Together with the query and time complexities of the (1Â±Îµ)-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 self-reducible problems in the polynomial-time regime by Jerrum, Valiant and Vazirani [Jerrum et al., 1986].
关键词：Graph sampling; Graph algorithms; Sublinear-time algorithms