NP-Completeness of Vertex Cover with Proof

# NP-Completeness of Vertex Cover with Proof Video Lecture - Theory of Computation - Computer Science Engineering (CSE)

## FAQs on NP-Completeness of Vertex Cover with Proof Video Lecture - Theory of Computation - Computer Science Engineering (CSE)

 1. What is the definition of NP-Completeness?
Ans. NP-Completeness refers to a category of computational problems that are classified as being among the most difficult problems in computer science. A problem is said to be NP-Complete if it belongs to the class of NP (nondeterministic polynomial time) problems and every other problem in NP can be polynomially reduced to it.
 2. How can we prove that a problem is NP-Complete?
Ans. To prove that a problem is NP-Complete, we need to show two things: first, that the problem belongs to the class NP, meaning that a proposed solution can be verified in polynomial time; and second, that the problem is NP-hard, meaning that every problem in NP can be reduced to it in polynomial time.
 3. What is the Vertex Cover problem?
Ans. The Vertex Cover problem is a well-known problem in graph theory, which involves finding the smallest possible set of vertices that covers all the edges in a given graph. In other words, it seeks to identify the minimum number of vertices that need to be included in a vertex cover, such that every edge in the graph is incident to at least one of these vertices.
 4. How can we prove that Vertex Cover is NP-Complete?
Ans. To prove that Vertex Cover is NP-Complete, we need to show that it belongs to the class NP and that it is NP-hard. The NP-Completeness proof typically involves reducing a known NP-Complete problem, such as the Boolean Satisfiability problem, to the Vertex Cover problem. By demonstrating that the Vertex Cover problem can be transformed into the known NP-Complete problem, we establish its NP-Completeness.
 5. What are the practical implications of Vertex Cover's NP-Completeness?
Ans. The NP-Completeness of the Vertex Cover problem implies that finding an exact solution for large graphs is computationally intractable. As a result, many algorithms for Vertex Cover focus on finding approximate solutions or heuristic approaches that provide reasonably good solutions in a reasonable amount of time. This has practical implications in various fields, such as network design, social network analysis, and resource allocation, where identifying a small set of crucial elements is essential.

## Theory of Computation

18 videos|56 docs|44 tests

## Theory of Computation

18 videos|56 docs|44 tests

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