Semirandom Planted Clique and the Restricted Isometry Property

We give a simple, greedy <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(n^{\omega+0.5})=O(n^{2.872})$</tex> - 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k\geq O(\sqrt{n}\log^{2}n)$</tex>. In the model, the edges touching the vertices in the planted k-clique are drawn independently with probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p=1/2$</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\log^{2}n)$</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n^{O(1/\varepsilon}$</tex>) -time algorithm to list-decode planted cliques of size <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k\geq\tilde{O}(n^{1/2+\varepsilon})$</tex>. In particular, the guarantee trivializes to quasi-polynomial time if the planted clique is of size <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O (\sqrt{n}$</tex> poly log <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex>). 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.