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

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.