Long-range frustration in Minimal Vertex Cover Problem on random graphs

TL;DR

The study introduces a percolation-based long-range frustration (LRF) framework to analyze the Minimal Vertex Cover problem on random graphs with general degree distributions. By formulating LRF within a cavity (Bethe-Peierls) approach, it derives self-consistent equations for coarse-grained vertex states and two cavity probabilities, enabling estimation of the MVC energy density $x$ solely from the degree distribution. Across ER, diluted regular, and scale-free graphs, the LRF predictions remain closer to survey propagation-guided decimation (SPD) results than to GLR-based analytics, particularly after the LRF percolation threshold $c^{*} = e$ is exceeded, where a nontrivial fraction of unfrozen vertices becomes type-I. The findings demonstrate that LRF captures essential long-range correlations that sculpt the energy landscape and ground-state properties of MVC, while reducing to GLR theory in the absence of long-range effects. This framework offers a general, degree-distribution-only description of MVC ground states on sparse graphs and suggests paths for refinement through more detailed unfrozen-state distributions.

Abstract

A vertex cover on a graph is a set of vertices in which each edge of the graph is adjacent to at least one vertex in the set. The Minimal Vertex Cover (MVC) Problem concerns finding vertex covers with a smallest cardinality. The MVC problem is a typical computationally hard problem among combinatorial optimization on graphs, for which both developing fast algorithms to find solution configurations on graph instances and constructing an analytical theory to estimate their ground-state properties prove to be difficult tasks. Here, by considering the long-range frustration (LRF) among MVC configurations and formulating it into a theoretical framework of a percolation model, we analytically estimate the energy density of MVCs on sparse random graphs only with their degree distributions. We test our framework on some typical random graph models. We show that, when there is a percolation of LRF effect in a graph, our predictions of energy densities are slightly higher than those from a hybrid algorithm of greedy leaf removal (GLR) procedure and survey propagation-guided decimation algorithm on graph instances, and there are still clearly closer to the results from the hybrid algorithm than an analytical theory based on GLR procedure, which ignores LRF effect and underestimates energy densities. Our results show that LRF is a proper mechanism in the formation of complex energy landscape of MVC problem and a theoretical framework of LRF helps to characterize its ground-state properties.

Figures (9)

• Figure 1: A diagram of MVC problem and vertex categories. (a) shows a small graph with $7$ vertices and $7$ edges. (b) and (c) show two MVC configurations, in which covered vertices are denoted as shaded circles, and uncovered vertices are in empty circles. (d) shows vertex categories with signs based on the two MVC configurations, in which a vertex with $0$ inside is frozen as being uncovered, a vertex with $\ast$ inside is in an unfrozen state, and a vertex with $1$ inside is frozen as being covered.
• Figure 2: Fixed point analysis and marginal probabilities of LRF theory of MVC problem on infinitely large ER random graphs. (a) shows the function $g(r_0, r_{\rm g})$ with different $c$. All the fixed points can be read from the intersection between $y = g(r_0, r_{\rm g})$ and $y = r_{\rm g}$. (b) shows marginal probabilities $(R_0, R_{\ast}, R_{\rm g}, R_1)$.
• Figure 3: Energy density of MVC problem on ER random graphs. We show here results from four methods: a hybrid algorithm of GLR procedure and BPD algorithm (GLR+BPD) on graph instances with a vertex size $N = 10^5$ with $\beta = 10$, a hybrid algorithm of GLR procedure and SPD algorithm (GLR+SPD) on graph instances with a vertex size $N = 10^5$ with $y = 3.1$, the framework of LRF theory (LRF) in the main text on infinitely large graphs with $\Delta c = 0.001$, and a theory based on GLR procedure (GLR-based) on infinitely large graphs.
• Figure 4: Marginal probabilities of LRF theory of MVC problem on infinitely large diluted RR graphs. (a-b) show marginal probabilities $(R_0, R_{\ast}, R_{\rm g}, R_1\}$ in the case of $K = 10$ and $K = 6$, respectively.
• Figure 5: Energy density of MVC problem on diluted RR graphs. (a)-(d) show energy densities from four methods in the case of $K = \{10, 8, 6, 4\}$, respectively. Each subgraph generally follows the format and the parameters in Figure \ref{['fig:er-e']}. For the GLR+SPD algorithm, we set $y = 3$. For the LRF theory, we set $\Delta \rho = 0.001$.