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We introduce a planted vertex cover problem on regular random graphs andstudy it by the cavity method of statistical mechanics. Different fromconventional Ising models, the equilibrium ferromagnetic phase transition ofthis binary-spin two-body interaction system is discontinuous, as theparamagnetic phase is separated from the ferromagnetic phase by an extensivefree energy barrier. The free energy landscape can be distinguished into threedifferent types depending on the two degree parameters of the planted graph.The critical inverse temperatures at which the paramagnetic phase becomeslocally unstable towards the ferromagnetic phase ($\beta_{\textrm{pf}}$) andtowards 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 threelandscapes. A locally stable anti-ferromagnetic phase emerges in the freeenergy landscape if $\beta_{\textrm{pf}} < \beta_{\textrm{pg}}$. When exploringthe free energy landscape by stochastic local search dynamics, we find that inagreement with our theoretical prediction, the first-passage time from theparamagnetic phase to the ferromagnetic phase is nonmonotonic with the inversetemperature. The potential relevance of the planted vertex cover model tosupercooled glass-forming liquids is briefly discussed.

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