How do you prove set cover is NP-complete?

Provide a collection C of subsets of size k, we can iterate over each element in the subsets of collection and mark the elements in X which are covered. At the end, no elements should be uncovered in X. This takes polynomial time with respect to the number of subsets in X. Hence, Set Cover is in NP.

How do you determine NP completeness of a problem?

The idea is to take a known NP-Complete problem and reduce it to L. If polynomial time reduction is possible, we can prove that L is NP-Complete by transitivity of reduction (If a NP-Complete problem is reducible to L in polynomial time, then all problems are reducible to L in polynomial time).

Is minimum set cover NP-complete?

In vertex cover, each vertex (i.e., set) covers the adjacent edges (i.e., elements); in set cover, each set can cover an arbitrary set of elements. Unfortunately, MIN-SET-COVER is NP-hard (i.e., NP-complete as a decision problem).

Is clique and set cover problem NP-complete?

Since VERTEX-COVER can be reduced to CLIQUE in polynomial time, CLIQUE ∈ NP and VERTEX-COVER is NP-Complete, CLIQUE is also NP-Complete.

Is the set cover problem NP-complete?

Thus, it can be verified that the set cover problem is NP-Complete using the following propositions:

How to prove that a problem is NP-complete?

Since an NP-Complete problem, by definition, is a problem which is both in NP and NP-hard, the proof for the statement that a problem is NP-Complete consists of two parts: All other problems in NP class can be polynomial-time reducible to that.

Is the set cover problem a complexity problem?

The set cover problem is a classical question in combinatorics, computer science and complexity theory. It is one of Karp’s 21 NP-complete problems shown to be NP-complete in 1972. It is a problem “whose study has led to the development of fundamental techniques for the entire field” of approximation algorithms.

Is the clique problem NP-complete or NP-hard?

For this, we consider the Independent Set problem, which is NP-Complete (and hence NP-Hard ). Every instance of the independent set problem consisting of the graph G (V, E) and an integer K can be converted to the required graph G’ (V’, E’) and K’ of the Clique problem.