Statistical mechanics of the minimum vertex cover problem in stochastic block models

TL;DR

This paper analyzes how mesoscopic community structures in stochastic block models influence the computational difficulty of the minimum vertex cover problem, revealing phase transitions based on intra- and inter-community degrees.

Contribution

It extends the understanding of Min-VC difficulty by incorporating stochastic block models and identifies conditions where problem complexity changes due to community structure.

Findings

Difficulty arises when $c_{in} + c_{out} > e$

Search becomes easier when $c_{out}$ is sufficiently larger than $c_{in}$

Experimental results support the cavity method predictions

Abstract

The minimum vertex cover (Min-VC) problem is a well-known NP-hard problem. Earlier studies illustrate that the problem defined over the Erd\"{o}s-R\'{e}nyi random graph with a mean degree $c$ exhibits computational difficulty in searching the Min-VC set above a critical point $c = e = 2.718 \ldots$. Here, we address how this difficulty is influenced by the mesoscopic structures of graphs. For this, we evaluate the critical condition of difficulty for the stochastic block model. We perform a detailed examination of the specific cases of two equal-size communities characterized by in- and out- degrees, which are denoted by $c_{\rm in}$ and $c_{\rm out}$, respectively. Our analysis based on the cavity method indicates that the solution search becomes difficult when $c_{\rm in }+c_{\rm out} > e$, but becomes easy again when $c_{\text{out}}$ is sufficiently larger than $c_{\mathrm{in}}$ in the region $c_{\rm out}>e$. Experiments based on various search algorithms support the theoretical prediction.