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- NP Hard and NP-Complete Classes
- NP-HARD AND NP-COMPLETE PROBLEMS
- NP-Completeness
- NP-HARD AND NP-COMPLETE PROBLEMS

In computational complexity theory , a problem is NP-complete when:. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines , a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a deterministic algorithm to check a single solution, or for a nondeterministic Turing machine to perform the whole search.

More precisely, each input to the problem should be associated with a set of solutions of polynomial length, whose validity can be tested quickly in polynomial time , [1] such that the output for any input is "yes" if the solution set is non-empty and "no" if it is empty. The complexity class of problems of this form is called NP , an abbreviation for "nondeterministic polynomial time". A problem is said to be NP-hard if everything in NP can be transformed in polynomial time into it even though it may not be in NP.

The NP-complete problems represent the hardest problems in NP. If any NP-complete problem has a polynomial time algorithm, all problems in NP do. Although a solution to an NP-complete problem can be verified "quickly", there is no known way to find a solution quickly. That is, the time required to solve the problem using any currently known algorithm increases rapidly as the size of the problem grows. As a consequence, determining whether it is possible to solve these problems quickly, called the P versus NP problem , is one of the fundamental unsolved problems in computer science today.

While a method for computing the solutions to NP-complete problems quickly remains undiscovered, computer scientists and programmers still frequently encounter NP-complete problems. NP-complete problems are often addressed by using heuristic methods and approximation algorithms. NP-complete problems are in NP , the set of all decision problems whose solutions can be verified in polynomial time; NP may be equivalently defined as the set of decision problems that can be solved in polynomial time on a non-deterministic Turing machine.

A problem p in NP is NP-complete if every other problem in NP can be transformed or reduced into p in polynomial time. It is not known whether every problem in NP can be quickly solved—this is called the P versus NP problem. But if any NP-complete problem can be solved quickly, then every problem in NP can, because the definition of an NP-complete problem states that every problem in NP must be quickly reducible to every NP-complete problem that is, it can be reduced in polynomial time.

Because of this, it is often said that NP-complete problems are harder or more difficult than NP problems in general. Note that a problem satisfying condition 2 is said to be NP-hard , whether or not it satisfies condition 1. The concept of NP-completeness was introduced in see Cook—Levin theorem , though the term NP-complete was introduced later.

At the STOC conference, there was a fierce debate between the computer scientists about whether NP-complete problems could be solved in polynomial time on a deterministic Turing machine. John Hopcroft brought everyone at the conference to a consensus that the question of whether NP-complete problems are solvable in polynomial time should be put off to be solved at some later date, since nobody had any formal proofs for their claims one way or the other.

Nobody has yet been able to determine conclusively whether NP-complete problems are in fact solvable in polynomial time, making this one of the great unsolved problems of mathematics. In , Richard Karp proved that several other problems were also NP-complete see Karp's 21 NP-complete problems ; thus there is a class of NP-complete problems besides the Boolean satisfiability problem.

Since the original results, thousands of other problems have been shown to be NP-complete by reductions from other problems previously shown to be NP-complete; many of these problems are collected in Garey and Johnson's book Computers and Intractability: A Guide to the Theory of NP-Completeness. An interesting example is the graph isomorphism problem , the graph theory problem of determining whether a graph isomorphism exists between two graphs.

Two graphs are isomorphic if one can be transformed into the other simply by renaming vertices. Consider these two problems:. The Subgraph Isomorphism problem is NP-complete. This is an example of a problem that is thought to be hard , but is not thought to be NP-complete.

The easiest way to prove that some new problem is NP-complete is first to prove that it is in NP, and then to reduce some known NP-complete problem to it. Therefore, it is useful to know a variety of NP-complete problems. The list below contains some well-known problems that are NP-complete when expressed as decision problems. To the right is a diagram of some of the problems and the reductions typically used to prove their NP-completeness.

In this diagram, problems are reduced from bottom to top. Note that this diagram is misleading as a description of the mathematical relationship between these problems, as there exists a polynomial-time reduction between any two NP-complete problems; but it indicates where demonstrating this polynomial-time reduction has been easiest.

There is often only a small difference between a problem in P and an NP-complete problem. For example, the 3-satisfiability problem, a restriction of the boolean satisfiability problem, remains NP-complete, whereas the slightly more restricted 2-satisfiability problem is in P specifically, NL-complete , and the slightly more general max.

Determining whether a graph can be colored with 2 colors is in P, but with 3 colors is NP-complete, even when restricted to planar graphs. Determining if a graph is a cycle or is bipartite is very easy in L , but finding a maximum bipartite or a maximum cycle subgraph is NP-complete.

A solution of the knapsack problem within any fixed percentage of the optimal solution can be computed in polynomial time, but finding the optimal solution is NP-complete. At present, all known algorithms for NP-complete problems require time that is superpolynomial in the input size, and it is unknown whether there are any faster algorithms.

The following techniques can be applied to solve computational problems in general, and they often give rise to substantially faster algorithms:.

Each vertex is a variable, edges are drawn between variables which are being used at the same time, and colors indicate the register assigned to each variable. Because most RISC machines have a fairly large number of general-purpose registers, even a heuristic approach is effective for this application. In the definition of NP-complete given above, the term reduction was used in the technical meaning of a polynomial-time many-one reduction. Another type of reduction is polynomial-time Turing reduction.

This contrasts with many-one reducibility, which has the restriction that the program can only call the subroutine once, and the return value of the subroutine must be the return value of the program. If one defines the analogue to NP-complete with Turing reductions instead of many-one reductions, the resulting set of problems won't be smaller than NP-complete; it is an open question whether it will be any larger. Another type of reduction that is also often used to define NP-completeness is the logarithmic-space many-one reduction which is a many-one reduction that can be computed with only a logarithmic amount of space.

Since every computation that can be done in logarithmic space can also be done in polynomial time it follows that if there is a logarithmic-space many-one reduction then there is also a polynomial-time many-one reduction. This type of reduction is more refined than the more usual polynomial-time many-one reductions and it allows us to distinguish more classes such as P-complete.

Whether under these types of reductions the definition of NP-complete changes is still an open problem. All currently known NP-complete problems are NP-complete under log space reductions. All currently known NP-complete problems remain NP-complete even under much weaker reductions. He reports that they introduced the change in the galley proofs for the book from "polynomially-complete" , in accordance with the results of a poll he had conducted of the theoretical computer science community.

The following misconceptions are frequent. Viewing a decision problem as a formal language in some fixed encoding, the set NPC of all NP-complete problems is not closed under:. From Wikipedia, the free encyclopedia. Complexity class. This article may be confusing or unclear to readers. Please help clarify the article. There might be a discussion about this on the talk page.

July Learn how and when to remove this template message. Main article: List of NP-complete problems. Boolean satisfiability problem SAT Knapsack problem Hamiltonian path problem Travelling salesman problem decision version Subgraph isomorphism problem Subset sum problem Clique problem Vertex cover problem Independent set problem Dominating set problem Graph coloring problem.

North Holland. Handbook of Theoretical Computer Science. Victor Klee ed. A Series of Books in the Mathematical Sciences. San Francisco, Calif. Freeman and Co. Journal of Computer and System Sciences. Computational Complexity. Garey, M. New York: W. This book is a classic, developing the theory, then cataloguing many NP-Complete problems. Cook, S. Dunne, P. COMP, Dept. Retrieved Crescenzi, P. KTH, Stockholm. Dahlke, K.

Math Reference Project. Karlsson, R. Archived from the original PDF on April 19, Sun, H. Information Security Laboratory, Dept. Archived from the original PPT on Jiang, J. Cormen, T. Sipser, M. Introduction to the Theory of Computation.

PWS Publishing. Papadimitriou, C. Addison Wesley. Bern, Marshall

Combinatorial Optimization — Eureka, You Shrink! We discuss fast exponential time solutions for NP-complete problems. We survey known results and approaches, we provide pointers to the literature, and we discuss several open problems in this area. The list of discussed NP-complete problems includes the travelling salesman problem, scheduling under precedence constraints, satisfiability, knapsack, graph coloring, independent sets in graphs, bandwidth of a graph, and many more. Unable to display preview. Download preview PDF. Skip to main content.

This is an on-line textbook on heuristic algorithms. It provides a nontechnical introduction to P-NP, its rich history, and its algorithmic implications for everything we do with computers and beyond. In this informative and entertaining book,. Book Site. Click here for details.

NP-Hard and NP-Complete Problems. Aims: • To describe SAT, a very important problem in complexity theory;. • To describe two more classes of problems: the.

Thus if we can solve L in polynomial time, we can solve all NP problems in polynomial time. If any NP-complete problem is solvable in polynomial time, then every NP-Complete problem is also solvable in polynomial time. Conversely, if we can prove that any NP-Complete problem cannot be solved in polynomial time, every NP-Complete problem cannot be solvable in polynomial time.

This is an on-line textbook on heuristic algorithms. It provides a nontechnical introduction to P-NP, its rich history, and its algorithmic implications for everything we do with computers and beyond. In this informative and entertaining book,. Book Site. Click here to find out.

Join Stack Overflow to learn, share knowledge, and build your career. Connect and share knowledge within a single location that is structured and easy to search. I am aware of many resources all over the web. I'd like to read your explanations, and the reason is they might be different from what's out there, or there is something that I'm not aware of. I assume that you are looking for intuitive definitions, since the technical definitions require quite some time to understand.

*In computational complexity theory , a problem is NP-complete when:.*

A problem is NP-hard if all problems in NP are polynomial time reducible to it, even though it may not be in NP itself. If a polynomial time algorithm exists for any of these problems, all problems in NP would be polynomial time solvable. These problems are called NP-complete. The phenomenon of NP-completeness is important for both theoretical and practical reasons. If a language satisfies the second property, but not necessarily the first one, the language B is known as NP-Hard.

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- Вы же учились в колледжах. Ну, кто-нибудь. Разница между ураном и плутонием. Ответа не последовало.

И конечно… ТРАНСТЕКСТ. Компьютер висел уже почти двадцать часов. Она, разумеется, знала, что были и другие программы, над которыми он работал так долго, программы, создать которые было куда легче, чем нераскрываемый алгоритм. Вирусы. Холод пронзил все ее тело.

Да нет, сэр, - попыталась она сгладить неловкость. - Не в этом дело… - Да в этом. - Он все еще посмеивался. - Дэвид Беккер хороший малый. Не упусти .

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## Somerville B.

– An optimization problem is one which asks,. “What is the optimal solution to problem X?” – Examples: • Knapsack. • Fractional Knapsack. • Minimum.

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Basic concepts We are concerned with distinction between the problems that can be solved by polynomial time algorithm and problems for which no polynomial time algorithm is known.

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