# What are Hitting sets?

## What are Hitting sets?

A hitting set for a collection of sets is a set that has a non- empty intersection with each set in the collection; the hitting set problem is to find a hitting set of minimum cardinality.

## Is hitting set NP hard?

Hitting Set is NP-Hard: In order to prove Hitting Set is NP-Hard, we will perform a reduction from which vertex cover problem can be reduced to the Hitting Set problem.

Is vertex cover a NP?

The minimum vertex cover problem is the optimization problem of finding a smallest vertex cover in a given graph. The vertex cover problem is an NP-complete problem: it was one of Karp’s 21 NP-complete problems. It is often used in computational complexity theory as a starting point for NP-hardness proofs.

### Is set cover in P?

Since our contruction takes polynomial time, and we have shown that Set Cover is in NP, we can conclude that Set Cover is NP-Complete.

### Is independent set NP?

The independent set decision problem is NP-complete, and hence it is not believed that there is an efficient algorithm for solving it. The maximum independent set problem is NP-hard and it is also hard to approximate.

What is the ρ approximation?

ρ-approximation algorithm (algorithmic technique) Definition: An approximation algorithm guaranteed to find a solution at most (or at least, as appropriate) ρ times the optimum. The ratio ρ is the performance ratio or relative performance guarantee of the algorithm.

## Is knapsack NP-complete?

Theorem 1 Knapsack is NP-complete. Proof: First of all, Knapsack is NP. The proof is the set S of items that are chosen and the verification process is to compute ∑i∈S si and ∑i∈S vi, which takes polynomial time in the size of input.

## Is Edge cover NP-complete?

Computing total edge covers. We now consider the t-Total Edge Cover problem. For this problem becomes the Edge Cover problem, which has long been known to be solvable in polynomial time . The problem is NP-complete for each fixed 2 ≤ t ≤ k [9, Theorem 3].

Why Is Set cover NP-complete?

Theorem: Set Cover is NP-Complete. Proof: First, we argue that Set Cover is in NP, since given a collection of sets C, a certifier can efficiently check that C indeed contains at most k elements, and that the union of all sets listed in C does include all elements from the ground set U.

### Is K independent set NP-complete?

Independent Set is NP-complete. k-Independent Set is in P for every k∈ℕ.

### What is C approximation algorithm?

We say that A is a c-approximation for P if for every input x of P: OPT(x) ≤ A(x) ≤ c · OPT(x) 2. If P is a maximization problem. We say that A is a c-approximation for P if for every input x of P: OPT(x) c ≤ A(x) ≤ OPT(x) In both cases, we call c the approximation ratio of A.

What is P N approximation algorithm?

If an algorithm reaches an approximation ratio of P(n), then we call it a P(n)-approximation algorithm. For a maximization problem, 0< C < C*, and the ratio of C*/C gives the factor by which the cost of an optimal solution is larger than the cost of the approximate algorithm.

## What is k-approximation of k- hitting set?

In computer science, k-approximation of k-hitting set is an approximation algorithm for weighted hitting set. The input is a collection S of subsets of some universe T and a mapping W from T to non-negative numbers called the weights of the elements of T.

## What is k-hitting set?

The input is a collection S of subsets of some universe T and a mapping W from T to non-negative numbers called the weights of the elements of T. In k-hitting set the size of the sets in S cannot be larger than k.

Why does it return a hitting set when it exits?

It returns a hitting set, because when the loop exits, all sets in S contain a tight element from T, and the set of these tight elements are returned. . This follows from the invariant property.