A Fixed Parameter Tractable Approach for Solving the Vertex Cover Problem in Polynomial Time Complexity

The Minimum Vertex Cover problem, a classical NP-complete problem, presents significant challenges for exact solution on large graphs. Fixed-Parameter Tractability (FPT) offers a powerful paradigm to address such problems by exploiting a parameter of the input, typically related to the size of the desired solution. This paper presents an implementation and empirical evaluation of an FPT algorithm for the Minimum Vertex Cover problem parameterized by the size of the vertex cover, $k$. The algorithm utilizes a branching strategy based on selecting adjacent vertices and recursively solving subproblems on a reduced graph. We describe the algorithmic approach, implementation details in Python, and present experimental results comparing its performance against the SageMath computational system. The results demonstrate that the FPT implementation achieves significant performance improvements for instances with large numbers of vertices ($n$) but relatively small values of the parameter ($k$), aligning with theoretical FPT complexity guarantees. We also discuss potential optimizations that could further improve the algorithm's performance, particularly concerning the branching factor.