Research on Solution Space of Bipartite Graph Vertex-Cover by Maximum Matchings

TL;DR

This paper analyzes the solution space of Vertex-Cover problems on bipartite graphs, providing exact algorithms, statistical insights, and exploring the complexity and structure of solutions, including the emergence of unfrozen cores and entropy calculations.

Contribution

It introduces an exact solution space expression algorithm based on Kőnig's theorem and extends analysis to bipartite core graphs, enhancing understanding of solution space complexity.

Findings

Solution space expression algorithm aligns with statistical results until unfrozen core emergence.

Entropy of solutions is calculated using clustering entropy and cycle simplification.

Insights into solution space structure aid in solving NP-complete problems and finding Kőnig-Egerváry subgraphs.

Abstract

Some rigorous results and statistics of the solution space of Vertex-Covers on bipartite graphs are given in this paper. Based on the $K\ddot{o}nig$'s theorem, an exact solution space expression algorithm is proposed and statistical analysis of the nodes' states is provided. The statistical results fit well with the algorithmic results until the emergence of the unfrozen core, which makes the fluctuation of statistical quantities and causes the replica symmetric breaking in the solutions. Besides, the entropy of bipartite Vertex-Cover solutions is calculated with the clustering entropy using a cycle simplification technique for the unfrozen core. Furthermore, as generalization of bipartite graphs, bipartite core graph is proposed, the solution space of which can also be easily determined; and based on these results, how to generate a $K\ddot{o}nig-Egerv\acute{a}ry$ subgraph is studied by a growth process of adding edges. The investigation of solution space of bipartite graph Vertex-Cover provides intensive understanding and some insights on the solution space complexity, and will produce benefit for finding maximal $K\ddot{o}nig-Egerv\acute{a}ry$ subgraphs, solving general graph Vertex-Cover and recognizing the intrinsic hardness of NP-complete problems.