File Name: np hard and np complete problems .zip
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Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs and how to get involved. Authors: Bas Lodewijks. Comments: 14 pages, 6 figures Subjects: Data Structures and Algorithms cs. DS ; Statistical Mechanics cond-mat.
In computational complexity theory , a problem is NP-complete when:. 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 ,  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.
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. Example for the first group is ordered searching its time complexity is O log n time complexity of sorting is O n log n. The second group is made up of problems whose known algorithms are non polynomial. Here we do is show that many of the problems for which there are no polynomial time algorithms are computationally related These are given the names NP hard and NP complete. A problem that is NP complete has the property that it can be solved in polynomial time iff all other NP complete problem can be solved in polynomial time If an NP hard problem can be solved in polynomial time ,then all NP complete problem can be solved in polynomial time. All NP-complete problems are NP-hard. The result of every operation is uniquely determined.
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 ,  such that the output for any input is "yes" if the solution set is non-empty and "no" if it is empty.
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. If a problem is proved to be NPC, there is no need to waste time on trying to find an efficient algorithm for it. Instead, we can focus on design approximation algorithm.
NP-hard: The class of problems to which every NP problem reduces. NP-complete (NPC): the class of problems which are NP-hard and belong to NP. NP-.
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