# Semirandom Planted Clique and the Restricted Isometry Property

CoRR（2024）

Abstract

We give a simple, greedy O(n^ω+0.5)=O(n^2.872)-time algorithm to
list-decode planted cliques in a semirandom model introduced in [CSV17]
(following [FK01]) that succeeds whenever the size of the planted clique is
k≥ O(√(n)log^2 n). In the model, the edges touching the vertices in
the planted k-clique are drawn independently with probability p=1/2 while
the edges not touching the planted clique are chosen by an adversary in
response to the random choices. Our result shows that the computational
threshold in the semirandom setting is within a O(log^2 n) factor of the
information-theoretic one [Ste17] thus resolving an open question of
Steinhardt. This threshold also essentially matches the conjectured
computational threshold for the well-studied special case of fully random
planted clique.
All previous algorithms [CSV17, MMT20, BKS23] in this model are based on
rather sophisticated rounding algorithms for entropy-constrained semidefinite
programming relaxations and their sum-of-squares strengthenings and the best
known guarantee is a n^O(1/ϵ)-time algorithm to list-decode planted
cliques of size k ≥Õ(n^1/2+ϵ). In particular, the
guarantee trivializes to quasi-polynomial time if the planted clique is of size
O(√(n)polylog n). Our algorithm achieves an almost
optimal guarantee with a surprisingly simple greedy algorithm.
The prior state-of-the-art algorithmic result above is based on a reduction
to certifying bounds on the size of unbalanced bicliques in random graphs –
closely related to certifying the restricted isometry property (RIP) of certain
random matrices and known to be hard in the low-degree polynomial model. Our
key idea is a new approach that relies on the truth of – but not efficient
certificates for – RIP of a new class of matrices built from the input graphs.

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