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

TL;DR

This work introduces a planted vertex cover model on regular random graphs and analyzes its thermodynamic landscape with the cavity method, revealing a discontinuous DS-to-CP transition and an intervening MP barrier that encodes the free-energy landscape. By deriving simplified BP fixed-point equations for two groups of vertices and performing a local stability analysis, the authors classify three landscape types according to the degrees $K$ and $K_{\text{ba}}$, and compute the critical inverse temperatures $\beta_{\textrm{pf}}$ and $\beta_{\textrm{pg}}$. They demonstrate a nonmonotonic mean first-passage time for escaping the paramagnetic phase, with an optimal temperature $\beta_{opt}$ in certain ensembles, and provide extensive numerical simulations that corroborate the theory. The results link planted graph optimization to concepts in glassy dynamics, spin glasses, and supercooled liquids, and suggest potential uses as two-body interaction benchmarks for quantum optimization and as models of crystallization and kinetic constraints in complex systems. The paper also highlights special graph ensembles where $\beta_{\textrm{pf}}=\beta_{\textrm{pg}}$, offering insight into the interplay between energetic and entropic barriers in inference problems.

Abstract

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 ($β_{\textrm{pf}}$) and towards spin glass phases ($β_{\textrm{pg}}$) satisfy $β_{\textrm{pf}} > β_{\textrm{pg}}$, $β_{\textrm{pf}} < β_{\textrm{pg}}$ and $β_{\textrm{pf}} = β_{\textrm{pg}}$, respectively, in these three landscapes. A locally stable anti-ferromagnetic phase emerges in the free energy landscape if $β_{\textrm{pf}} < β_{\textrm{pg}}$. 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.

Figures (11)

• Figure 1: Free energy landscape of the planted vertex cover problem. (a) Regular random graph containing two groups of vertices and free of edges within group $A$. (b)-(d) Schematic plots of free energy density $f$ versus the fraction $\rho_{\textrm{b}}$ of occupied vertices in group $B$ at different fixed inverse temperatures ($\beta_1 < \beta_F < \beta_2 < \beta_{\textrm{pf}} < \beta_3$). The three phases DS, MP and CP are respectively indicated by the solid, dotted, and dashed thick lines. The equilibrium discontinuous phase transition occurs at $\beta_{F}$. (b) DS phase is locally stable. (c) DS phase is locally unstable when inverse temperature exceeds $\beta_{\textrm{pf}}$. (d) DS and MP phases touch at $\beta_{\textrm{pf}}$, and at $\beta_3 > \beta_{\textrm{pf}}$ both the DS and MP phases (indicated by small circles) are saddle points and they can evolve to the CP phase without passing through each other.
• Figure 2: The mapping $g(\gamma)$ as defined by Eq. (\ref{['eq:ggamma']}) for $K=10$ and $K_{\textrm{ba}}=6$. There is always a trivial root $\gamma=1$ for the equation $\gamma = g(\gamma)$. When $\beta > 0.3351$ two additional non-negative roots appear, one of them always being less than unity. The two curves are obtained at $\beta = 0.20$ (dashed) and $\beta=0.45$ (solid). (a) $g(\gamma)$ versus $\gamma$. (b) $g(\gamma) - \gamma$ versus $\gamma$.
• Figure 3: Theoretical results on the planted regular random graph ensemble with degree parameters $K=10$, $K_{\textrm{ba}} = 5$. Results corresponding to the disordered symmetric (DS), microcanonical polarized (MP), and canonical polarized (CP) fixed points are distinguished by different line and color types. (a) Energy density $\rho$ versus inverse temperature $\beta$. Inset shows the difference of energy density $\Delta \rho = \rho_{\textrm{MP}}-\rho_{\textrm{DS}}$ between the MP and DS phases. (b) Magnetization $m$ versus $\beta$. (c) Free energy density $f$ versus $\beta$. Inset is a magnified view of the region $\beta \in (1.1, 1.3)$. (d) Entropy density $s$ versus $\rho$. Inset is a magnified view of the region $\rho \in (0.765, 0.805)$.
• Figure 4: The rescaled free energy density difference $\beta \Delta f$ (a) and entropy density difference $\Delta s$ (b) between the intermediate MP phase and the paramagnetic DS phase, for the planted regular random graph ensemble with degree parameters $K=10$, $K_{\textrm{ba}}=5$. Vertical dashed lines mark the spin glass dynamic phase transition point ($\beta_d$ and $\rho_d$), vertical dotted lines mark the optimal escaping point ($\beta_{opt}$ and $\rho_{opt}$).
• Figure 5: Simulation results obtained on single planted regular random graph instances with degree parameters $K=10$ and $K_{B A} = 5$. (a) Mean energy density $\rho$ at fixed inverse temperature $\beta$ (canonical Monte Carlo, circle) and mean inverse temperature $\beta$ at fixed energy density $\rho$ (microcanonical Monte Carlo, diamond), for a single graph containing $N=2100$ vertices. The curves are theoretical predictions as in Fig. \ref{['fig:K10v5v5:a']}, and the vertical and horizontal dashed lines indicate the equilibrium phase transition points. (b) Median waiting time $\tau$ needed to make the transition from the paramagnetic DS phase to the ferromagnetic CP phase at inverse temperature $\beta$, on a single graph of size $N$, whose value is $768$ (circle) and $1536$ (diamond). The escaping time $\tau$ is rescaled by $\tau_0 = e^{0.0053\, N}$. The vertical dotted line marks the predicted optimal inverse temperature $\beta_{opt} = 2.5580$, the vertical dashed line locates the spin glass transition inverse temperature $\beta_d = 2.9560$.