Improved polynomial-time algorithms for detecting and recovering planted Θ(√n)-cliques
Abstract
In the planted clique problem, one observes either an Erdős--Rényi graph on n vertices or such a graph with a clique added to k = k(n) vertices, and seeks to detect or recover the clique. It is widely believed that k = Θ(√n) is the smallest clique size for which polynomial-time algorithms exist for these tasks. We develop new algorithms in this regime using color-coding to estimate signed subgraph counts, further accelerated with fast matrix multiplication. We first show that, for each t ≥ 1, for c(t) a constant associated to the order of growth of the number of connected graphs of treewidth at most t, cliques of size k = λ√n planted in a random location with λ> 1 / √(c(t)) can be detected and recovered in time nt + 1 + o(1). For instance, since c(1) = e, this recovers by counting signed trees the performance of the Õ(n2)-time message-passing algorithm of Deshpande--Montanari (2015) that succeeds when λ> 1 / √e ≈ 0.6066. For t ≥ 3, the exact value of c(t) is not known, but lower bounds on it give a hierarchy of slower polynomial-time algorithms that succeed for smaller λ. We further show that the above algorithm for t = 2 can be implemented in time nω+ o(1) for ω the constant of square matrix multiplication and succeeds when λ> 0.3320; under the folklore conjecture that ω= 2, this runs in the nearly-linear time of the algorithm of Deshpande--Montanari while finding smaller cliques. Second, we show that the above algorithm for t = 1 can be combined with the boosting scheme of Alon--Krivelevich--Sudakov (1998) using rectangular matrix multiplication, giving improved runtimes for smaller λ. Taken together, our results achieve the best known tradeoff between runtime and signal strength λ.
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BibTeX
@misc{kunisky2026improved,
title = {{Improved polynomial-time algorithms for detecting and recovering planted $Θ(\sqrtn)$-cliques}},
author = {Dmitriy Kunisky and Songtao Mao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24780},
eprint = {2609.24780},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24780},
}
RIS
TY - UNPB
TI - Improved polynomial-time algorithms for detecting and recovering planted Θ(√n)-cliques
AU - Dmitriy Kunisky
AU - Songtao Mao
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24780
UR - https://arxiv.org/abs/2609.24780
AB - In the planted clique problem, one observes either an Erdős--Rényi graph on n vertices or such a graph with a clique added to k = k(n) vertices, and seeks to detect or recover the clique. It is widely believed that k = Θ(√n) is the smallest clique size for which polynomial-time algorithms exist for these tasks. We develop new algorithms in this regime using color-coding to estimate signed subgraph counts, further accelerated with fast matrix multiplication. We first show that, for each t ≥ 1, for c(t) a constant associated to the order of growth of the number of connected graphs of treewidth at most t, cliques of size k = λ√n planted in a random location with λ> 1 / √(c(t)) can be detected and recovered in time n^t + 1 + o(1). For instance, since c(1) = e, this recovers by counting signed trees the performance of the Õ(n^2)-time message-passing algorithm of Deshpande--Montanari (2015) that succeeds when λ> 1 / √e ≈ 0.6066. For t ≥ 3, the exact value of c(t) is not known, but lower bounds on it give a hierarchy of slower polynomial-time algorithms that succeed for smaller λ. We further show that the above algorithm for t = 2 can be implemented in time n^ω+ o(1) for ω the constant of square matrix multiplication and succeeds when λ> 0.3320; under the folklore conjecture that ω= 2, this runs in the nearly-linear time of the algorithm of Deshpande--Montanari while finding smaller cliques. Second, we show that the above algorithm for t = 1 can be combined with the boosting scheme of Alon--Krivelevich--Sudakov (1998) using rectangular matrix multiplication, giving improved runtimes for smaller λ. Taken together, our results achieve the best known tradeoff between runtime and signal strength λ.
ER -