- Trending Categories
Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies
- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who
Prove that the polynomial time reduction is from the Clique problem to the Vertex Cover problem
Vertex cover is a subset of vertices that covers all the edges in a graph. It is used to determine whether a given graph has a 3SAT to vertex cover.
Clique is called a subset of vertices that are all directly connected. It determines whether a clique of size k exists in a graph.
To prove − Vertex cover can be reduced to clique.
Proof
Given a graph G=(V,E) and integer k.
Get its complement graph G'=(V,E').
Solve CLIQUE(G',|V|-k).
If there is a solution, return yes. Otherwise, it returns as no.
To prove this reduction, we need to show the following −
If there is a solution to VERTEX-COVER(G,k), then there must be a solution to CLIQUE(G',|V|-k) and vice versa.
Assume that G has a vertex cover V' ⊆ V , where |V |' = k. Then, for all u, v ε V , if (u, v) ε E, then u ε V' or v ε V' or both since the vertex cover must cover all edges.
The contrapositive is that for all u, v ε V, if u does not belong to V', then (u, v) ε/ E and thus (u, v) ε E.
For any pair of vertices that are both not in the vertex cover V' of G, there is an edge between them in G.
The union of all pairs of vertices that are all not in V' is simply V − V'. Thus, V − V' is a clique in G, by definition, and V − V' has size |V | − k.
This operation can be done in polynomial time. Since VERTEX-COVER can be reduced to CLIQUE in polynomial time, CLIQUE ε NP and VERTEX-COVER is NP-complete, CLIQUE is also NP-Complete.
2K+ Views
