# The Lov\'asz Theta Function for Random Regular Graphs and Community Detection in the Hard Regime

## TL;DR

This paper investigates the limitations of the Lovász theta function and sum-of-squares proofs in refuting k-colorability in random regular graphs, revealing computational hardness in certain regimes and providing bounds related to graph girth.

## Contribution

It establishes bounds on the degree for which the Lovász theta function can refute k-colorability, showing failure above the phase transition and linking to community detection hardness.

## Key findings

- Refutation fails above the k-colorability transition.
- Refutation fails below the Kesten-Stigum threshold.
- Provides explicit bounds on theta for regular graphs with given girth.

## Abstract

We derive upper and lower bounds on the degree $d$ for which the Lov\'asz $\vartheta$ function, or equivalently sum-of-squares proofs with degree two, can refute the existence of a $k$-coloring in random regular graphs $G_{n,d}$. We show that this type of refutation fails well above the $k$-colorability transition, and in particular everywhere below the Kesten-Stigum threshold. This is consistent with the conjecture that refuting $k$-colorability, or distinguishing $G_{n,d}$ from the planted coloring model, is hard in this region. Our results also apply to the disassortative case of the stochastic block model, adding evidence to the conjecture that there is a regime where community detection is computationally hard even though it is information-theoretically possible. Using orthogonal polynomials, we also provide explicit upper bounds on $\vartheta(\overline{G})$ for regular graphs of a given girth, which may be of independent interest.