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arXiv · 2212.05619

Algorithms approaching the threshold for semi-random planted clique

Rares-Darius Buhai·Pravesh K. Kothari·David Steurer
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Abstract

We design new polynomial-time algorithms for recovering planted cliques in the semi-random graph model introduced by Feige and Kilian 2001. The previous best algorithms for this model succeed if the planted clique has size at least $n^{2/3}$ in a graph with $n$ vertices (Mehta, Mckenzie, Trevisan 2019 and Charikar, Steinhardt, Valiant 2017). Our algorithms work for planted-clique sizes approaching $n^{1/2}$ -- the information-theoretic threshold in the semi-random model (Steinhardt 2017) and a conjectured computational threshold even in the easier fully-random model. This result comes close to resolving open questions by Feige 2019 and Steinhardt 2017. Our algorithms are based on higher constant degree sum-of-squares relaxation and rely on a new conceptual connection that translates certificates of upper bounds on biclique numbers in unbalanced bipartite Erdős--Rényi random graphs into algorithms for semi-random planted clique. The use of a higher-constant degree sum-of-squares is essential in our setting: we prove a lower bound on the basic SDP for certifying bicliques that shows that the basic SDP cannot succeed for planted cliques of size $k =o(n^{2/3})$. We also provide some evidence that the information-computation trade-off of our current algorithms may be inherent by proving an average-case lower bound for unbalanced bicliques in the low-degree-polynomials model.

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Rares-Darius Buhai, Pravesh K. Kothari, David Steurer. 2023-06-06. Algorithms approaching the threshold for semi-random planted clique. https://arxiv.org/abs/2212.05619

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