We give a simple, greedy $O(n^{\omega+0.5})=O(n^{2.872})$-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 $k\geq O(\sqrt{n} \log^2 n)$. 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 $n^{O(1/\epsilon)}$-time algorithm to list-decode planted cliques of size $k \geq \tilde{O}(n^{1/2+\epsilon})$. In particular, the guarantee trivializes to quasi-polynomial time if the planted clique is of size $O(\sqrt{n} \operatorname{polylog} n)$. 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.

翻译：我们给出一个简单贪心的$O(n^{\omega+0.5})=O(n^{2.872})$时间算法，用于在[CSV17]（承[FK01]）提出的半随机模型中对团进行列表解码，该算法在植入团规模$k\geq O(\sqrt{n} \log^2 n)$时成功。在该模型中，与植入的$k$-团相连的边以概率$p=1/2$独立生成，而未触及植入团的边则由对手根据随机选择结果自适应确定。我们的结果表明，半随机设置下的计算阈值与信息论阈值[Ste17]仅相差$O(\log^2 n)$因子，从而解决了Steinhardt提出的开放问题。该阈值也本质上匹配了充分随机植入团这一经典特例的猜想计算阈值。此前该模型中的所有算法[CSV17, MMT20, BKS23]均基于熵约束半定规划松弛及其平方和强化版本的复杂舍入算法，已知最优保证是$n^{O(1/\epsilon)}$时间算法可对规模$k \geq \tilde{O}(n^{1/2+\epsilon})$的植入团进行列表解码。特别地，当植入团规模为$O(\sqrt{n} \operatorname{polylog} n)$时，该保证退化为拟多项式时间。我们的算法通过一个惊人简单的贪心方法实现了近乎最优的保证。此前最先进的算法结果基于归约到随机图中非平衡双团规模的认证问题——该问题与随机矩阵的限制等距性质认证密切相关，且已知在低阶多项式模型下难以处理。我们的核心创新在于：不依赖高效证书，而是直接利用由输入图构造的新型矩阵族满足限制等距性质这一事实本身。