Detecting the solution space of vertex cover by mutual determinations and backbones

Abstract

To solve the combinatorial optimization problems especially the minimal Vertex-cover problem with high efficiency, is a significant task in theoretical computer science and many other subjects. Aiming at detecting the solution space of Vertex-cover, a new structure named mutualdetermination between unfrozen nodes is defined and discovered for arbitrary graphs, which results in the emergence of the strong correlations. Based on the backbones and mutual-determinations with node ranks by leaf removal, we propose an Mutual-determination and Backbone Evolution Algorithm to achieve the reduced solution graph, which provides a graphical expression of the solution space of Vertex-cover. By this algorithm, the whole solution space and detailed structures such as backbones can be obtained strictly when there is no leaf-removal core on the graph. Compared with the current algorithms, the Mutual-determination and Backbone Evolution Algorithm performs better than the replica symmetry ones but has a small gap higher than the replica symmetric breaking ones, and has a relatively small error for the exact results. The algorithm with the mutual-determination provides a new viewpoint to solve Vertex-cover, by which all detailed information of the ground/steady states can be shown in the reduced solution graph.

Research on the Solution Space of 2-SAT and Max-2-SAT

“…Similar as the analysis in the minimum vertexCover problem, we mainly concern with two structures on the solution space: backbones and mutualdeterminations [17].…”

“…Similar as the organization of solution space of Vertexcover [17], we define the reduced solution graph of Max-2-SAT, which is mainly based on the two important structures backbones and mutual-determination and the evolution of them. The reduced graph is based on the factor graph with the x=1 backbones yellow circles, x=0 backbones red circles, free nodes blue circles and the mutual-determination red squares.…”

“…In order to detect the structure of solution space, directly applying the analysis of [12] and [17], we add one new node (called i) to previous N-1 system (N-1 nodes), new node connects to other k nodes x1, x2, … , xk. We suppose k1 nodes free, k2 nodes frozen to 1, k3 nodes frozen to 0 in previous system, by which k1 + k2 +k3=k.…”

“…Following [17], in this paper, we focus on the solution space of Max-2-SAT. Backbones and mutual determination are discussed to understand the structure of solution space of 2-SAT and Max-2-sat problems.…”

“…Like spin glass approach, cavity method, phase transition ( [11], [12], [14] ). In [17], a new method is proposed to detect the whole solution space of the minimum vertex cover problem, and we want to generalize the corresponding analysis to 2-SAT and Max-2-SAT problem.…”