CS3016/02. Program graph theory 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.

Table of contents

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.

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)

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