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 Lovász 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 Lovász 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：图着色与团覆盖计算问题是组合优化领域的核心挑战。已知这两类问题均属于NP难问题，因此在最坏情况下是计算不可行的。一种求解此类问题近似解的著名方法是备受推崇的Lovász theta函数$\vartheta(G)$，该函数被定义为半定规划（SDP）的解，因而具备可计算性。本研究突破最坏情况分析的局限，致力于探究Lovász theta函数能否为具有潜在团覆盖结构（可能受噪声干扰）的随机实例恢复团覆盖。我们对此问题给出了肯定答案，并证明对于本文提出的植入团模型所生成的图，$\vartheta(G)$的SDP公式具有唯一解，能以高概率揭示底层的团覆盖结构。关键技术步骤是一个中间结果，其中我们基于恰当的稀疏性概念证明了恢复的确定性条件。