A Proposed Algorithm for Minimum Vertex Cover Problem and its Testing

TL;DR

This paper introduces a new algorithm for the NP-Complete minimum vertex cover problem, demonstrating high success rates on large random graphs and analyzing its computational complexity.

Contribution

The paper proposes a novel multi-stage algorithm for minimum vertex cover, with proven efficiency and high applicability on large random graphs, and suggests further stages for universal applicability.

Findings

Stage 1 yields a proved minimum vertex cover for 99.99% of tested graphs of order 16.

Stage 2 is applicable to all tested graphs, including those of order 32.

The algorithm runs in polynomial time, specifically $O(n^{5+logn})$ and $O(n^{3(5+logn)/2})$ for stages 1 and 2.

Abstract

The paper presents an algorithm for minimum vertex cover problem, which is an NP-Complete problem. The algorithm computes a minimum vertex cover of each input simple graph. Tested by the attached MATLAB programs, Stage 1 of the algorithm is applicable to, i.e., yields a proved minimum vertex cover for, about 99.99% of the tested 610,000 graphs of order 16 and 99.67% of the tested 1,200 graphs of order 32, and Stage 2 of the algorithm is applicable to all of the above tested graphs. All of the tested graphs are randomly generated graphs of random "edge density" or in other words, random probability of each edge. It is proved that Stage 1 and Stage 2 of the algorithm run in $O(n^{5+logn})$ and $O(n^{3(5+logn)/2})$ time respectively, where $n$ is the order of input graph. Because there is no theoretical proof yet that Stage 2 is applicable to all graphs, further stages of the algorithm are proposed, which are in a general form that is consistent with Stages 1 and 2.