The problems of computing graph colorings and clique covers are central challenges in combinatorial optimization. Both of these are known to be NP-hard, and thus computationally intractable in the worst-case instance. A prominent approach for computing approximate solutions to these problems is the celebrated Lov\'asz theta function $\vartheta(G)$, which is specified as the solution of a semidefinite program (SDP), and hence tractable to compute. In this work, we move beyond the worst-case analysis and set out to understand whether the Lov\'asz theta function recovers clique covers for random instances that have a latent clique cover structure, possibly obscured by noise. We answer this question in the affirmative and show that for graphs generated from the planted clique model we introduce in this work, the SDP formulation of $\vartheta(G)$ has a unique solution that reveals the underlying clique-cover structure with high-probability. The main technical step is an intermediate result where we prove a deterministic condition of recovery based on an appropriate notion of sparsity.

翻译：图着色与团覆盖问题的计算是组合优化中的核心挑战。这两个问题均已知为NP难问题，因此在最坏情形下计算上不可解。求解这些问题的近似解的一个著名方法是Lovász Theta函数$\vartheta(G)$，该函数被定义为半定规划（SDP）的解，因此是可计算的。在本工作中，我们超越最坏情形分析，旨在探究Lovász Theta函数是否能在具有潜在团覆盖结构（可能被噪声掩盖）的随机实例中恢复团覆盖。我们对这一问题给出肯定回答，并证明：对于本文提出的植入团模型生成的图，$\vartheta(G)$的SDP公式具有唯一解，该解以高概率揭示潜在团覆盖结构。主要技术步骤是一个中间结果：我们基于适当定义的稀疏性概念，证明了恢复的确定性条件。