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NP Complete

In your journey to enrich your understanding of computer science, delving into the complex world of 'NP Complete' is essential. This compelling aspect of the theory of computation presents both challenge and fascination, as we will discover together in the forthcoming sections. Our exploration kicks off with a deciphering of this elusive term, setting the stage with its definition and historical context. It then extends further to distinguish between NP Hard and NP Complete, by reflecting on their key differences and similarities. Next, to deepen your comprehension, we take a metaphorical dive into the practical side of things by discussing the ways in which NP Complete problems manifest within the essence of computer science discipline. You will be presented with common examples of such problems as well as various strategies employed in approaching their solutions. Finally, we enrich your wisdom trove with a hands-on tutorial to prove NP Completeness, clarifying its daunting aspects by going through an easy-to-understand example problem. This holistic guide is dedicated to unravelling the intriguing concept of NP Complete, encouraging you to explore, learn, and contribute to this fascinating domain of computer science.

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Jetzt kostenlos anmeldenIn your journey to enrich your understanding of computer science, delving into the complex world of 'NP Complete' is essential. This compelling aspect of the theory of computation presents both challenge and fascination, as we will discover together in the forthcoming sections. Our exploration kicks off with a deciphering of this elusive term, setting the stage with its definition and historical context. It then extends further to distinguish between NP Hard and NP Complete, by reflecting on their key differences and similarities. Next, to deepen your comprehension, we take a metaphorical dive into the practical side of things by discussing the ways in which NP Complete problems manifest within the essence of computer science discipline. You will be presented with common examples of such problems as well as various strategies employed in approaching their solutions. Finally, we enrich your wisdom trove with a hands-on tutorial to prove NP Completeness, clarifying its daunting aspects by going through an easy-to-understand example problem. This holistic guide is dedicated to unravelling the intriguing concept of NP Complete, encouraging you to explore, learn, and contribute to this fascinating domain of computer science.

In the field of computer science, the term NP Complete (Non-deterministic Polynomial-time Complete) is often used. These are decision problems for which a 'yes' answer can be verified in polynomial time. However, there is yet no polynomial-time algorithm discovered that can either provide 'yes' or 'no' answers to them.

**Problem Statement**: A problem statement is a concise summary of an issue in computer science that needs to be addressed or resolved.

For example, if you have a list of cities and the distances between each pair of them, the problem of identifying the shortest possible route that covers all cities and returns back to the origin city, is known as the Travelling Salesman Problem (TSP). This problem is a classic NP Complete problem.

The concept of ‘time complexity’ forms the crux of NP Complete problems. Every algorithm requires a certain amount of time to run. This time is generally expressed as a function of the input size - referred to as the ‘time complexity’ of the algorithm. For problems classified as NP Complete, the time complexity increases much faster than the size of the input.

**NP Hard**: In computational complexity theory, an NP-hard (Non-deterministic Polynomial-time hard) problem is one that is at least as difficult as the hardest problems in NP. Essentially, any problem to which an NP Complete problem can be polynomially reduced is NP Hard.

So, how does one distinguish between an NP Hard and an NP Complete problem? Let's delve a bit deeper into their key differences and similarities:

- An NP Complete problem is a special type of NP Hard problem where the problem itself is in NP. If a problem is NP Hard but not in NP, it is simply an NP Hard problem, not NP Complete.
- An NP Complete problem has solutions that can be validated quickly, whereas this is not a requirement for an NP Hard problem.
- Given an NP Complete problem, one should be able to transform it into any other NP Complete problem in polynomial time.

As an illustration, consider the Chess problem, i.e., given a position in a chess game, seeing if the white player has a forced win. This problem is NP Hard but not NP Complete, because there is no efficient algorithm to verify a solution.

In fact, if you find a polynomial time algorithm to solve any NP Complete problem, then you’ve simultaneously discovered a polynomial time solution for all problems in NP! This is because every problem in NP reduces to every NP Complete problem. In this respect, the pursuit of solutions to NP Complete problems underpins much pivotal research in computer science.

**Travelling Salesman Problem (TSP)**: Given a set of cities and the distances between each pair, find the shortest possible tour that visits each city exactly once, returning to the starting city.**Knapsack Problem**: Given a set of items, each with a weight and a value, determine the number of each object to include in a bag, ensuring the weight does not exceed a particular limit while maximising the total value.**Vertex Cover Problem**: Given a graph and an integer k, the problem is to determine whether there exists a set of vertices of size k such that every edge of the graph is incident (connected to) to at least one vertex in the set.

**Brute Force**: This approach tries all possible solutions until it finds the best one. Though it guarantees an optimal solution, it is highly impractical for large problems.**Greedy Algorithms**: These algorithms make the locally optimal choice at each stage with the hope that these local solutions will lead to a global optimum. However, this is not always accurate for several NP Complete problems.**Dynamic Programming**: Commonly used for solving the knapsack problem, this technique breaks the problem into simpler subproblems and solves each one only once, storing their solutions using a memory-based data structure (array, table, etc.).**Backtracking**: This is a refined brute force approach which abandons a solution as soon as it determines that the solution cannot be improved upon any further.

As an example, a problem like the Travelling Salesman Problem (TSP) can be addressed using heuristic methods such as the Nearest Neighbour algorithm or 2-Opt algorithm, both of which aim to create a good approximation of the optimal tour.

The subsequent step after possessing these necessary pre-requisites is to tackle the process of proving a problem NP Complete. The proof involves two fundamental steps: showing that the problem is in NP, and then showing that it's NP Hard.

The TSP involves a salesman needing to travel through n cities and return back to his initial city, while ensuring the total distance travelled is as short as possible. This problem can be represented by a complete graph, with vertices representing the cities and the edge weights representing the distances between the cities. The aim is to find a Hamiltonian cycle - a cycle that visits every vertex just once and returns to the starting point - with the minimum weight.

NP Complete refers to decision problems in computer science for which a 'yes' answer can be verified in polynomial time, but there's no known polynomial-time algorithm that can provide 'yes' or 'no' answers.

The problem statement in computational theory refers to a question we're trying to answer; NP Complete problem statements are questions that may be easy to solve on a small scale but become increasingly difficult with a larger problem size.

A common NP Complete problem is the Travelling Salesman Problem (TSP), which involves finding the shortest possible route that covers all cities and returns back to the origin city.

American computer scientist Stephen Cook introduced the concept of NP completeness in 1971; this concept was further developed, and several other problems were identified as NP Complete by Richard Karp in 1972.

The time complexity of an algorithm, which expresses the amount of time an algorithm requires to run as a function of the input size, increases much faster than the input size for problems classified as NP Complete.

To prove a problem is NP-complete, you must demonstrate two things. Firstly, the problem is in NP (meaning a solution can be verified in polynomial time). Secondly, every problem in NP can be reduced to it in polynomial time, meaning if you had a fast algorithm solving this problem, you could solve all NP problems quickly. This reduction process typically involves showing that an already known NP-complete problem can be transformed into the problem you're trying to prove is NP-Complete.

No, factoring is not NP-complete. It is in the class of problems named NP-Intermediate, residing between P and NP-Complete. However, it's important to note that NP-Intermediacy is contingent on P not equalling NP - a question that remains unanswered in computer science.

NP-complete problems are a class of computational problems for which no efficient solution has been found, but if a solution were found, it could be verified efficiently. In other words, non-deterministic polynomial (NP) complete problems are problems whose solutions can be verified in polynomial time, and every NP problem can be reduced to them through a polynomial time transformation. Examples include the travelling salesman problem, the knapsack problem, and Boolean satisfiability problem. These problems are significant in computer science and mathematics due to their implications on computational complexity theory.

NP Complete stands for 'Nondeterministic Polynomial time Complete'. It is a complexity class in computational theory, referring to the hardest problems in NP (Nondeterministic Polynomial time), for which no efficient solutions exist. If any NP Complete problem has an efficient solution, all problems in NP would have efficient solutions. They are decision problems, which means their answers can be either 'yes' or 'no'.

Yes, all NP-Complete problems are considered hard. 'Hardness' in this context refers to computational complexity. Specifically, a problem being NP-Complete means that it is at least as 'hard' as any problem in NP, and that its solution can be checked quickly, but a fast solution algorithm is not known.

Flashcards in NP Complete15

Start learningWhat is the definition of an NP Complete problem in computer science?

An NP Complete problem is a decision problem for which a 'yes' answer can be verified in polynomial time, but no polynomial-time algorithm has been discovered that can either provide 'yes' or 'no' answers.

What is the historic context of NP Complete in computational theory?

NP Complete was introduced by American computer scientist Stephen Cook in 1971. It was further developed by Richard Karp in 1972, who demonstrated that several other problems were also NP Complete by identifying 'Karp's 21 NP-Complete problems'.

What is a classic example of an NP Complete problem?

The Travelling Salesman Problem (TSP), which is the problem of identifying the shortest possible route that covers all cities and returns to the origin city, is a classic example of an NP Complete problem.

What does the term 'NP Hard' mean in computational complexity theory?

NP Hard refers to problems that are at least as difficult as the hardest problems in NP. Essentially, any problem to which an NP Complete problem can be polynomially reduced is NP Hard.

What are the key differences between NP Hard and NP Complete problems?

An NP Complete problem is a special type of NP Hard problem that is in NP, and its solutions can be validated quickly. An NP Complete problem can be transformed into any other NP Complete problem in polynomial time. These are not requirements for an NP Hard problem.

What is the significance of NP Complete problems in the field of computer science?

NP Complete problems are key in understanding computational complexity, studying the efficiency of algorithms. Recognising a problem as NP Complete means an exact solution within a reasonable time might not be achievable, leading to heuristic or approximation algorithms.

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