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：我们提出了一种简单、贪心的 $O(n^{\omega+0.5})=O(n^{2.872})$ 时间算法，用于在 [CSV17]（沿袭 [FK01]）提出的半随机模型中列表解码植入团，该算法在植入团大小 $k\geq O(\sqrt{n} \log^2 n)$ 时均能成功。在该模型中，与植入的 $k$-团顶点相连的边以概率 $p=1/2$ 独立抽取，而不接触植入团的边则由对抗方根据随机选择的结果进行选定。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 提出的一个开放性问题。该阈值也基本匹配了被广泛研究的完全随机植入团特例中猜想中的计算阈值。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加强的复杂舍入算法，已知的最佳保证是一个 $n^{O(1/\epsilon)}$ 时间算法，用于列表解码大小 $k \geq \tilde{O}(n^{1/2+\epsilon})$ 的植入团。特别地，当植入团大小为 $O(\sqrt{n} \operatorname{polylog} n)$ 时，该保证退化为拟多项式时间。我们的算法通过一种出奇简单的贪心算法实现了近乎最优的保证。上述先前的最先进算法结果基于将问题归约为证明随机图中不平衡二分团大小的界——这与证明某些随机矩阵的受限等距性质（RIP）密切相关，并且在低次多项式模型中已知是困难的。我们的核心思想是一种新方法，它依赖于一类基于输入图构建的新矩阵的 RIP 性质的真实性，但不需要其高效验证。