Vertex Cover Problem in DAA

Learn the Vertex Cover Problem in DAA using Backtracking. Understand the problem statement, minimum vertex cover, algorithm, example, time complexity O(2^E), and applications in networking and optimization.

Introduction

The Vertex Cover Problem is an important graph problem.
In a given graph, the objective is to select a subset of vertices such that every edge in the graph is incident to at least one selected vertex.

Problem Statement

Given an undirected graph
G(V, E)

Objective:
To find a vertex cover C such that for every edge (u, v),

u ∈ C or v ∈ C

where C is the selected set of vertices.

Types of Vertex Cover

Vertex Cover – Any valid vertex cover
Minimum Vertex Cover – A vertex cover with the minimum number of vertices
(This problem is NP-Complete)

Example

Edges in the graph:
(1,2), (2,3), (3,4)

One possible vertex cover is:
{2, 3}

Vertex Cover Using Backtracking

For each edge, there are two choices:
- Select the first vertex of the edge
- Select the second vertex of the edge
While selecting vertices, it is checked whether the remaining edges are covered
If all edges are covered, a solution is obtained
If a wrong choice is made, the algorithm backtracks

State Space Tree

Each level of the tree represents one edge
Branches represent vertex selection or deselection
If an invalid path is found, the algorithm backtracks

Vertex Cover Algorithm (Backtracking)

VertexCover(edge_index)
{
if (all edges are covered)
print solution
else if (edge_index < E)
{
(u, v) = edge[edge_index]

// Select u
include u
VertexCover(edge_index + 1)
exclude u // Backtrack

// Select v
include v
VertexCover(edge_index + 1)
exclude v // Backtrack
}
}

Time Complexity

Worst Case Time Complexity: O(2ᴱ)
Because there are two choices for each edge

Space Complexity

O(V) – To store the selected vertex set
O(E) – Recursive function calls

Advantages of the Vertex Cover Problem

Helps in understanding graph optimization problems
Important for the study of NP-Complete problems
Useful in network security and resource allocation

Limitations

Requires a large amount of time for large graphs
Finding an exact solution is computationally expensive

Applications of the Vertex Cover Problem-

Network Monitoring
Facility Location Problems
Compiler Design (Register Allocation)
Bioinformatics

Conclusion

The Vertex Cover Problem is an important NP-Complete problem that demonstrates the
limitations of algorithmic power.
The Backtracking method provides an exact solution, but it becomes impractical for large inputs due to high time complexity.

Some More:

Vertex Cover Problem in DAA