Graphical Representation and Hierarchical Decomposition Mechanism for Vertex-Cover Solution Space
Abstract
In this paper, solution space organization of minimum vertex-cover problem is deeply investigated using the K\"{o}nig-Eg\'{e}rvary (KE) graph and theorem, in which a hierarchical decomposition mechanism named KE-layer structure of general graphs is proposed to reveal the complexity of vertex-cover. An algorithm to verify the KE graph is given by the solution space expression of vertex-cover, and the relation between multi-layer KE graphs and maximal matching is illustrated and proved. Furthermore, a framework to calculate the KE-layer number and approximate the minimal vertex-cover is provided, with different strategies of switching nodes and counting energy. The phase transition phenomenon between different KE-layers are studied with the transition points located, and searching of vertex-cover got by this strategy presents comparable advantage against several other methods. The graphical representation and hierarchical decomposition provide a new perspective to illustrate the intrinsic complexity for large-scale graphs/systems recognition.
Cite
@article{arxiv.1912.08559,
title = {Graphical Representation and Hierarchical Decomposition Mechanism for Vertex-Cover Solution Space},
author = {Wei Wei and Xiangnan Feng and Jiannan Wang and Xue Liu and Zhiming Zheng},
journal= {arXiv preprint arXiv:1912.08559},
year = {2021}
}
Comments
20 pages, 6 figures