Approximately Counting and Sampling Hamiltonian Motifs in Sublinear Time
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Author(s)
Eden, Talya
Levi, Reut
Ron, Dana
Rubinfeld, Ronitt
Date Issued
June 15, 2025
Publisher
ACM|Proceedings of the 57th Annual ACM Symposium on Theory of Computing
Citation
Talya Eden, Reut Levi, Dana Ron, and Ronitt Rubinfeld. 2025. Approximately Counting and Sampling Hamiltonian Motifs in Sublinear Time. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC '25). Association for Computing Machinery, New York, NY, USA, 1043–1054.
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Final published version
Abstract
Counting small subgraphs, referred to as motifs, in large graphs is
a fundamental task in graph analysis, extensively studied across
various contexts and computational models. In the sublinear-time
regime, the relaxed problem of approximate counting has been
explored within two prominent query frameworks: the standard
model, which permits degree, neighbor, and pair queries, and the
strictly more powerful augmented model, which additionally allows
for uniform edge sampling. Currently, in the standard model, (optimal) results have been established only for approximately counting
edges, stars, and cliques, all of which have a radius of one. This
contrasts sharply with the state of affairs in the augmented model,
where algorithmic results (some of which are optimal) are known
for any input motif, leading to a disparity which we term the “scope
gap" between the two models.
In this work, we make significant progress in bridging this gap.
Our approach draws inspiration from recent advancements in the
augmented model and utilizes a framework centered on counting
by uniform sampling, thus allowing us to establish new results in
the standard model and simplify on previous results.
In particular, our first, and main, contribution is a new algorithm
in the standard model for approximately counting any Hamiltonian
motif in sublinear time, where the complexity of the algorithm
is the sum of two terms. One term equals the complexity of the
known algorithms by Assadi, Kapralov, and Khanna (ITCS 2019)
and Fichtenberger and Peng (ICALP 2020) in the (strictly stronger)
augmented model and the other is an additional, necessary, additive
overhead.
Our second contribution is a variant of our algorithm that enables nearly uniform sampling of these motifs, a capability previously limited in the standard model to edges and cliques. Our
third contribution is to introduce even simpler algorithms for stars
and cliques by exploiting their radius-one property. As a result, we
simplify all previously known algorithms in the standard model for
stars (Gonen, Ron, Shavitt (SODA 2010)), triangles (Eden, Levi, Ron Seshadhri (FOCS 2015)) and cliques (Eden, Ron, Seshadri (STOC
2018)).
Description
STOC ’25, Prague, Czechia
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1145/3717823.3718160
