Statistics and algorithms on the organization and quantitative analysis of minimal vertex covers.

Counting the solution number of combinational optimization problems is an important topic in the study of computational complexity, especially research on the #P-complete complexity class can provide profound understanding of the hardness of the NP-complete complexity class. In this paper, we first investigate some topological structures of the unfrozen subgraph of Vertex-Cover unfrozen vertices based on its solution space expression, including the degree distribution and the relationship of their vertices with edges, which indicates that giant component of unfrozen vertices appears simultaneously with the leaf-removal core and reveals the organization of the further-step replica symmetry breaking phenomenon. Furthermore, a solution number counting algorithm of Vertex-Cover is proposed, which is exact when the graphs have no leaf-removal core and the unfrozen subgraphs are (almost) forests. To have a general view of the solution space of Vertex-Cover, the marginal probability distributions of the unfrozen vertices are investigated to recognize the efficiency of survey propagation algorithm. And, the algorithm is applied on the scale-free graphs to see different evolution characteristics of the solution space. Thus, detecting the graph expression of the solution space is an alternative and meaningful way to study the hardness of NP-complete and #P-complete problems, and appropriate algorithm design can help to achieve better approximations of solving combinational optimization problems and corresponding counting problems.