Determining the Solution Space of Vertex-Cover by Interactions and Backbones

TL;DR

This paper introduces a new approach to analyze the solution space of the Vertex-cover problem using interactions and backbones, leading to an exact and efficient algorithm for certain graph classes.

Contribution

It proposes a novel Interaction and Backbone Evolution Algorithm that accurately characterizes the solution space of Vertex-cover and improves solving efficiency for random graphs.

Findings

The algorithm can obtain the entire solution space when no leaf-removal core exists.

It demonstrates high accuracy and performance on random graphs with high average degrees.

Interactions reveal long-range correlations and frustration phenomena in the solution space.

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 interaction between nodes is defined and discovered for random graph, which results in the emergence of the frustration and long-range correlation phenomenon. Based on the backbones and interactions with a node adding process, we propose an Interaction and Backbone Evolution Algorithm to achieve the reduced solution graph, which has a direct correspondence to the solution space of Vertex-cover. By this algorithm, the whole solution space can be obtained strictly when there is no leaf-removal core on the graph and the odd cycles of unfrozen nodes bring great obstacles to its efficiency. Besides, this algorithm possesses favorable exactness and has good performance on random instances even with high average degrees. The interaction with the algorithm provides a new viewpoint to solve Vertex-cover, which will have a wide range of applications to different types of graphs, better usage of which can lower the computational complexity for solving Vertex-cover.