Lab 02: Polynomial-Time Reductions Among Clique, Vertex Cover, and Independent Set Problems

To understand and implement polynomial-time reductions among the Clique, Vertex Cover, and Independent Set problems, which are fundamental in computational complexity and NP-completeness.

Introduction

Graph theory plays a crucial role in computer science, particularly in optimization and computational complexity. Three well-known NP-complete problems in graph theory are:

1. Clique Problem: Finding the largest complete subgraph (clique) in a given graph.
2. Vertex Cover Problem: Finding the smallest set of vertices that covers all edges.
3. Independent Set Problem: Finding the largest set of mutually non-adjacent vertices.

These problems are interrelated, and polynomial-time reductions among them help demonstrate their computational hardness. Understanding these reductions is essential in complexity theory and real-world applications such as network security and bioinformatics.

Some must-read references for this topic are:

• Independent Set, Clique, and Vertex Cover (Lecturer: Abrahim Ladha) 

• Google Dork: site:edu “clique OR vertex OR cover OR independent set” 

Reductions Among Problems

1. Clique to Independent Set Reduction
   • Given a graph $G = (V, E)$, the complement graph $\bar{G}$ is constructed.
   • A k-clique in $G$ corresponds to a k-independent set in $\bar{G}$.
2. Independent Set to Vertex Cover Reduction
   • Given a graph $G = (V, E)$, a set $S \subseteq V$ is independent if and only if $V - S$ is a vertex cover.
   • Finding the largest independent set is equivalent to finding the smallest vertex cover.
3. Vertex Cover to Clique Reduction
   • Given a graph $G = (V, E)$, the complement graph $\bar{G}$ is considered.
   • A k-vertex cover in $G$ corresponds to a |V| - k clique in $\bar{G}$.

Algorithm Implementations

1. Constructing Graph Complement

import networkx as nx
 
def graph_complement(graph):
    return nx.complement(graph)

2. Clique to Independent Set Reduction

def clique_to_independent_set(graph, k):
    complement_graph = graph_complement(graph)
    return nx.algorithms.approximation.maximum_independent_set(complement_graph)

3. Independent Set to Vertex Cover Reduction

def independent_set_to_vertex_cover(graph, independent_set):
    return set(graph.nodes) - set(independent_set)

4. Vertex Cover to Clique Reduction

def vertex_cover_to_clique(graph, vertex_cover):
    complement_graph = graph_complement(graph)
    return set(complement_graph.nodes) - set(vertex_cover)

Example Usage

G = nx.erdos_renyi_graph(6, 0.5)  # Generate a random graph
independent_set = clique_to_independent_set(G, 3)
vertex_cover = independent_set_to_vertex_cover(G, independent_set)
clique = vertex_cover_to_clique(G, vertex_cover)
print("Independent Set:", independent_set)
print("Vertex Cover:", vertex_cover)
print("Clique:", clique)

Visualization

import networkx as nx
import matplotlib.pyplot as plt
import random
 
def find_independent_set(graph):
    return nx.algorithms.approximation.maximum_independent_set(graph)
 
def find_vertex_cover(graph):
    return nx.algorithms.approximation.min_weighted_vertex_cover(graph)
 
def find_max_clique(graph):
    return nx.algorithms.approximation.clique.max_clique(graph)
 
def compute_complementary_vertex_cover(graph, independent_set):
    return set(graph.nodes) - set(independent_set)
 
def visualize_graph(G, highlighted_nodes, title, color):
    plt.figure(figsize=(8, 6))
    pos = nx.spring_layout(G)
    nx.draw(G, pos, with_labels=True, node_color='lightgray', edge_color='gray')
    if highlighted_nodes:
        nx.draw_networkx_nodes(G, pos, nodelist=highlighted_nodes, node_color=color, label=title)
    plt.title(title)
    plt.legend()
    plt.show()
 
# Generate a random graph
random.seed(42)
G = nx.erdos_renyi_graph(8, 0.4)
 
# Find different sets
independent_set = find_independent_set(G)
vertex_cover = find_vertex_cover(G)
computed_vertex_cover = compute_complementary_vertex_cover(G, independent_set)
clique = find_max_clique(G)
 
# Print results
print("Independent Set:", independent_set)
print("Vertex Cover (Approximation):", vertex_cover)
print("Vertex Cover (Computed as Complement):", computed_vertex_cover)
print("Clique:", clique)
 
# Visualize each graph separately
visualize_graph(G, independent_set, "Independent Set", "red")
visualize_graph(G, vertex_cover, "Vertex Cover (Approximation)", "blue")
visualize_graph(G, computed_vertex_cover, "Vertex Cover (Complement)", "purple")
visualize_graph(G, clique, "Clique", "green")

Time and Space Complexity Analysis

Applications

1. Network Security: Identifying crucial network nodes.
2. Social Networks: Finding tightly connected groups (Cliques).
3. Computational Biology: Protein interaction networks.
4. Scheduling Problems: Assigning tasks without conflicts.

Conclusion

Polynomial-time reductions among Clique, Vertex Cover, and Independent Set demonstrate their computational equivalence and NP-completeness. Understanding these reductions is vital for algorithm design and complexity analysis.

References

1. Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C.
   Introduction to Algorithms, 3rd/4th Edition, MIT Press.

2. P vs. NP and the Computational Complexity Zoo (2014) 

3. (Warm-Up) Implement divide-and-conquer algorithms for binary search, merge sort, and quick sort.