Planted vertex cover problem on regular random graphs and nonmonotonic temperature-dependence in the supercooled region

We introduce a planted vertex cover problem on regular random graphs and study it by the cavity method of statistical mechanics. Different from conventional Ising models, the equilibrium ferromagnetic phase transition of this binary-spin two-body interaction system is discontinuous, as the paramagnetic phase is separated from the ferromagnetic phase by an extensive free energy barrier. The free energy landscape can be distinguished into three different types depending on the two degree parameters of the planted graph. The critical inverse temperatures at which the paramagnetic phase becomes locally unstable towards the ferromagnetic phase ($\beta_{\textrm{pf}}$) and towards spin glass phases ($\beta_{\textrm{pg}}$) satisfy $\beta_{\textrm{pf}} > \beta_{\textrm{pg}}$, $\beta_{\textrm{pf}} < \beta_{\textrm{pg}}$ and $\beta_{\textrm{pf}} = \beta_{\textrm{pg}}$, respectively, in these three landscapes. A locally stable anti-ferromagnetic phase emerges in the free energy landscape if $\beta_{\textrm{pf}} < \beta_{\textrm{pg}}$. When exploring the free energy landscape by stochastic local search dynamics, we find that in agreement with our theoretical prediction, the first-passage time from the paramagnetic phase to the ferromagnetic phase is nonmonotonic with the inverse temperature. The potential relevance of the planted vertex cover model to supercooled glass-forming liquids is briefly discussed.