We design new polynomial-time algorithms for recovering planted cliques in the semi-random graph model introduced by Feige and Kilian 2001. The previous best algorithms for this model succeed if the planted clique has size at least $n^{2/3}$ in a graph with $n$ vertices (Mehta, Mckenzie, Trevisan 2019 and Charikar, Steinhardt, Valiant 2017). Our algorithms work for planted-clique sizes approaching $n^{1/2}$ -- the information-theoretic threshold in the semi-random model (Steinhardt 2017) and a conjectured computational threshold even in the easier fully-random model. This result comes close to resolving open questions by Feige 2019 and Steinhardt 2017. Our algorithms are based on higher constant degree sum-of-squares relaxation and rely on a new conceptual connection that translates certificates of upper bounds on biclique numbers in unbalanced bipartite Erd\H{o}s--R\'enyi random graphs into algorithms for semi-random planted clique. The use of a higher-constant degree sum-of-squares is essential in our setting: we prove a lower bound on the basic SDP for certifying bicliques that shows that the basic SDP cannot succeed for planted cliques of size $k =o(n^{2/3})$. 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-polynomials model.

翻译：我们设计了新的多项式时间算法，用于恢复Feige和Kilian（2001）提出的半随机图模型中的植入团。该模型此前的最优算法要求植入团在$n$个顶点的图中大小至少为$n^{2/3}$（Mehta, McKenzie, Trevisan 2019; Charikar, Steinhardt, Valiant 2017）。我们的算法可处理大小趋近于$n^{1/2}$的植入团——这是半随机模型的信息论阈值（Steinhardt 2017），甚至在更简单的完全随机模型中也被认为是计算上的阈值。这一结果接近解决了Feige（2019）和Steinhardt（2017）提出的公开问题。我们的算法基于更高常数次数的平方和松弛，并依赖于一个新的概念性联系：将非平衡二部Erdős–Rényi随机图中双团数上界的证书，转化为半随机植入团的算法。更高常数次数的平方和在本研究中至关重要：我们证明了用于认证双团的基本半定规划（SDP）的下界，表明基本SDP无法成功处理大小为$k = o(n^{2/3})$的植入团。此外，我们通过低阶多项式模型中的平均情况非平衡双团下界，提供了当前算法信息-计算权衡可能具有固有性的证据。