The Lovász Theta Function for Recovering Planted Clique Covers and Graph Colorings

TL;DR

The paper investigates whether the Lovász theta function, computable via a semidefinite program, can efficiently recover planted clique covers in random graphs with latent clique structure obscured by noise. It proves that for graphs generated under a planted clique cover model with inter-clique edge probability $p$ below a threshold $c$, the SDP has a unique optimal solution revealing the latent clique cover with high probability, by constructing dual certificates and proving an extremality property. A key contribution is an incoherence-type analysis and a deterministic $c$-SCC condition that guarantees exact recovery, extended to planted clique covers via a probabilistic argument. Numerical experiments compare Lovász theta to several SDP/ILP-based baselines, showing strong performance of theta up to moderate noise and revealing phase-transition-like behavior as problem size grows. The results advance beyond-worst-case understanding of colorings and clique covers, connecting SDP relaxations with latent combinatorial structure and offering practical implications for clustering and community detection under noise.

Abstract

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.

Figures (4)

• Figure 1: A planted clique instance with $8$ cliques of size $8$ each, and with parameter $p=0.2$.
• Figure 2: Comparison of Lovász theta function with other methods.
• Figure 3: Comparison of the clique covering number with Lovász theta for random clique cover model (left). Comparison of time taken for ILP solver to compute clique covering number with time taken for SDP solver to compute Lovász theta function (right). The number of simplex iterations taken by ILP solver is included as a reference.
• Figure 4: Probability of correctly recovering a planted clique instance with increasing clique size and increasing number of cliques. The black curves correspond to strong recovery while the red curves correspond to weak recovery. The thickness of the lines increase with the problem parameters $n$ and $k^\star$.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof