Is the knapsack problem NP-complete?

Is the knapsack problem NP-complete?

Computational complexity The decision problem form of the knapsack problem (Can a value of at least V be achieved without exceeding the weight W?) is NP-complete, thus there is no known algorithm both correct and fast (polynomial-time) in all cases. There is a pseudo-polynomial time algorithm using dynamic programming.

How do you prove knapsack problem is 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.

What is NP-complete with example?

NP-complete problem, any of a class of computational problems for which no efficient solution algorithm has been found. Many significant computer-science problems belong to this class—e.g., the traveling salesman problem, satisfiability problems, and graph-covering problems.

Is partition problem NP-complete?

Although the partition problem is NP-complete, there is a pseudo-polynomial time dynamic programming solution, and there are heuristics that solve the problem in many instances, either optimally or approximately. The optimization version is NP-hard, but can be solved efficiently in practice.

Is vertex cover NP complete?

Its decision version, the vertex cover problem, was one of Karp’s 21 NP-complete problems and is therefore a classical NP-complete problem in computational complexity theory.

How many steps are required to prove that a decision problem is NP complete?

13. How many steps are required to prove that a decision problem is NP complete? Explanation: First, the problem should be NP. Next, it should be proved that every problem in NP is reducible to the problem in question in polynomial time.

How do you solve NP-complete problems?

Solving NP-complete problems

1. Approximation: Instead of searching for an optimal solution, search for a solution that is at most a factor from an optimal one.
2. Randomization: Use randomness to get a faster average running time, and allow the algorithm to fail with some small probability.

What is NP-hard problem with example?

Examples. An example of an NP-hard problem is the decision subset sum problem: given a set of integers, does any non-empty subset of them add up to zero? That is a decision problem and happens to be NP-complete.

How do you prove a problem is NP-hard?

To prove that problem A is NP-hard, reduce a known NP-hard problem to A. In other words, to prove that your problem is hard, you need to describe an ecient algorithm to solve a dierent problem, which you already know is hard, using an hypothetical ecient algorithm for your problem as a black-box subroutine.

Will P vs NP ever be solved?

Although one-way functions have never been formally proven to exist, most mathematicians believe that they do, and a proof of their existence would be a much stronger statement than P ≠ NP. Thus it is unlikely that natural proofs alone can resolve P = NP.

Can NP problems be solved in polynomial time?

NP stands for Non-deterministic Polynomial time. This means that the problem can be solved in Polynomial time using a Non-deterministic Turing machine (like a regular Turing machine but also including a non-deterministic “choice” function). Basically, a solution has to be testable in poly time.

Is Sudoku an NP complete problem?

Mathematical context The general problem of solving Sudoku puzzles on n2×n2 grids of n×n blocks is known to be NP-complete. The Sudoku graph has 81 vertices, one vertex for each cell. The vertices are labeled with ordered pairs (x, y), where x and y are integers between 1 and 9.

How do you solve vertex cover problems?

Given an undirected graph, the vertex cover problem is to find minimum size vertex cover. The following are some examples. Vertex Cover Problem is a known NP Complete problem, i.e., there is no polynomial-time solution for this unless P = NP. There are approximate polynomial-time algorithms to solve the problem though.

What is clique in algorithm?

By convention, in algorithm analysis, the number of vertices in the graph is denoted by n and the number of edges is denoted by m. A clique in a graph G is a complete subgraph of G. That is, it is a subset K of the vertices such that every two vertices in K are the two endpoints of an edge in G.

What is the size of a minimum vertex cover in KN?

Hence the minimum size of a vertex cover can be 3. We can check in O(E + V) time if a given subset of vertices is a vertex cover or not, using the following algorithm.

Is a dominating set a vertex cover?

Dominating Set: As with vertex cover, dominating set is an example of a graph covering problem. Here the condition is a little different, each vertex is adjacent to at least one member of the dominating set, as opposed to each edge being incident to at least one member of the vertex cover.

Is dominating set NP-complete?

A dominating set of size k exists in G if and only if a vertex cover of size k exists in G. Since we know that vertex cover is an NP-complete problem, Dominating Set is also NP-complete.

How do you find the vertex cover of a graph?

A vertex-cover of an undirected graph G = (V, E) is a subset of vertices V’ ⊆ V such that if edge (u, v) is an edge of G, then either u in V or v in V’ or both.

What is dominating set in graph theory?

In graph theory, a dominating set for a graph G = (V, E) is a subset D of V such that every vertex not in D is adjacent to at least one member of D. The domination number γ(G) is the number of vertices in a smallest dominating set for G.

What is matching in graph?

Given an undirected graph, a matching is a set of edges, such that no two edges share the same vertex. In other words, matching of a graph is a subgraph where each node of the subgraph has either zero or one edge incident to it. A vertex is said to be matched if an edge is incident to it, free otherwise.

How do you find the maximal independent set of a graph?

A maximum independent line set of ‘G’ with maximum number of edges is called a maximum independent line set of ‘G’. L3 is the maximum independent line set of G with maximum edges which are not the adjacent edges in graph and is denoted by β1 = 3. Line independent number (Matching number) = β1 = [n/2] α1 + β1 = n.

How do you prove NP complete?

We can solve Y in polynomial time: reduce it to X. Therefore, every problem in NP has a polytime algorithm and P = NP. then X is NP-complete. In other words, we can prove a new problem is NP-complete by reducing some other NP-complete problem to it.

How many NP-complete problems are there?

This list is in no way comprehensive (there are more than 3000 known NP-complete problems). Most of the problems in this list are taken from Garey and Johnson’s seminal book Computers and Intractability: A Guide to the Theory of NP-Completeness, and are here presented in the same order and organization.

What is the difference between NP-hard and NP-complete?

A non-deterministic Turing machine can solve NP-Complete problem in polynomial time….Difference between NP-Hard and NP-Complete:

NP-hard	NP-Complete
To solve this problem, do not have to be in NP .	To solve this problem, it must be both NP and NP-hard problems.
Do not have to be a Decision problem.	It is exclusively a Decision problem.