The Lov\'asz Theta Function for Recovering Planted Clique Covers and Graph Colorings

TL;DR

This paper demonstrates that the Lovász theta function, computed via semidefinite programming, can reliably recover planted clique covers and graph colorings in random graphs with latent structures, surpassing worst-case limitations.

Contribution

The work proves that the Lovász theta function can recover planted clique covers in random graphs, providing theoretical guarantees beyond worst-case scenarios.

Findings

Lovász theta function recovers clique covers in planted models

Unique SDP solution reveals underlying structure with high probability

Provides deterministic conditions for recovery based on sparsity

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 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.