Graphical Representation and Hierarchical Decomposition Mechanism for Vertex-Cover Solution Space

TL;DR

This paper investigates the solution space of the minimum vertex-cover problem using KE graphs, proposing a hierarchical decomposition mechanism and algorithms that improve understanding and approximation of vertex-covers in large graphs.

Contribution

It introduces the KE-layer structure for hierarchical decomposition of vertex-cover solutions and provides algorithms for verification, approximation, and analysis of phase transitions.

Findings

KE-layer structure reveals vertex-cover complexity

Proposed algorithms improve approximation of minimal vertex-cover

Phase transition points identified in KE-layer hierarchy

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.