Towards Optimal Output-Sensitive Clique Listing or: Listing Cliques from Smaller Cliques
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Author(s)
Dalirrooyfard, Mina
Mathialagan, Surya
Williams, Virginia Vassilevska
Xu, Yinzhan
Date Issued
June 10, 2024
Publisher
ACM|Proceedings of the 56th Annual ACM Symposium on Theory of Computing
Citation
Dalirrooyfard, Mina, Mathialagan, Surya, Williams, Virginia Vassilevska and Xu, Yinzhan. 2024. "Towards Optimal Output-Sensitive Clique Listing or: Listing Cliques from Smaller Cliques."
Version
Final published version
Abstract
We study the problem of nding and listing -cliques in an -edge,
-vertex graph, for constant ≥ 3. This is a fundamental problem
of both theoretical and practical importance.
Our rst contribution is an algorithmic framework for nding
-cliques that gives the rst improvement in 19 years over the
old runtimes for 4 and 5-clique nding, as a function of [Eisenbrand and Grandoni, TCS’04]. With the current bounds on matrix
multiplication, our algorithms run in (1.66) and (2.06) time,
respectively, for 4-clique and 5-clique nding.
Our main contribution is an output-sensitive algorithm for listing -cliques, for any constant ≥ 3. We complement the algorithm
with tight lower bounds based on standard ne-grained assumptions. Previously, the only known conditionally optimal outputsensitive algorithms were for the case of 3-cliques given by Björklund, Pagh, Vassilevska W. and Zwick [ICALP’14]. If the matrix
multiplication exponent is 2, and if the number of -cliques is
large enough, the running time of our algorithms is
˜
min{
1
−2
1−
2
(−2)
,
2
−1
1−
2
(−1) }
,
and this is tight under the Exact--Clique Hypothesis. This running
time naturally extends the running time obtained by Björklund,
Pagh, Vassilevska W. and Zwick for = 3.
Our framework is very general in that it gives -clique listing
algorithms whose running times can be measured in terms of the
number of ℓ-cliques Δℓ
in the graph for any 1 ≤ ℓ < . This
generalizes the typical parameterization in terms of (the number
of 1-cliques) and (the number of 2-cliques).
If is 2, and if the size of the output, Δ
, is su ciently large,
then for every ℓ < , the running time of our algorithm for listing
-cliques is
˜
Δ
2
ℓ(−ℓ)
ℓ
Δ
1−
2
(−ℓ)
Description
STOC ’24, June 24–28, 2024, Vancouver, BC, Canada
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
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DOI of Published Version
https://doi.org/10.1145/3618260.3649663
