@inproceedings{da3daaeb3e504633ad5953e68f21556f,
title = "Algorithms Approaching the Threshold for Semi-random Planted Clique",
abstract = "We design new polynomial-time algorithms for recovering planted cliques in the semi-random graph model introduced by Feige and Kilian. The previous best algorithms for this model succeed if the planted clique has size at least n2/3 in a graph with n vertices. Our algorithms work for planted-clique sizes approaching n1/2-the information-theoretic threshold in the semi-random model and a conjectured computational threshold even in the easier fully-random model. This result comes close to resolving open questions by Feige and Steinhardt. To generate a graph in the semi-random planted-clique model, we first 1) plant a clique of size k in an n-vertex-graph with edge probability 1/2 and then adversarially add or delete an arbitrary number edges not touching the planted clique and delete any subset of edges going out of the planted clique. For every {"}>0, we give an nO(1/{"})-time algorithm that recovers a clique of size k in this model whenever k ≥ n1/2+{"}. In fact, our algorithm computes, with high probability, a list of about n/k cliques of size k that contains the planted clique. Our algorithms also extend to arbitrary edge probabilities p and improve on the previous best guarantee whenever p ≤ 1-n-0.001. Our algorithms rely on a new conceptual connection that translates certificates of upper bounds on biclique numbers in unbalanced bipartite-random graphs into algorithms for semi-random planted clique. Analogous to the (conjecturally) optimal algorithms for the fully-random model, the previous best guarantees for semi-random planted clique correspond to spectral relaxations of biclique numbers based on eigenvalues of adjacency matrices. We construct an SDP lower bound that shows that the n2/3 threshold in prior works is an inherent limitation of these spectral relaxations. We go beyond this limitation by using higher-order sum-of-squares relaxations for biclique numbers. 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 polynomial model.",
keywords = "planted clique, semi-random, semidefinite programming, sum-of-squares hierarchy",
author = "Buhai, {Rares Darius} and Kothari, {Pravesh K.} and David Steurer",
note = "Publisher Copyright: {\textcopyright} 2023 ACM.; 55th Annual ACM Symposium on Theory of Computing, STOC 2023 ; Conference date: 20-06-2023 Through 23-06-2023",
year = "2023",
month = jun,
day = "2",
doi = "10.1145/3564246.3585184",
language = "English (US)",
series = "Proceedings of the Annual ACM Symposium on Theory of Computing",
publisher = "Association for Computing Machinery",
pages = "1918--1926",
editor = "Barna Saha and Servedio, {Rocco A.}",
booktitle = "STOC 2023 - Proceedings of the 55th Annual ACM Symposium on Theory of Computing",
}