In computational complexity theory, **NP-hardness** (non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard problem is the subset sum problem.

A more precise specification is: a problem *H* is NP-hard when every problem *L* in NP can be reduced in polynomial time to *H*; that is, assuming a solution for *H* takes 1 unit time, *H*'s solution can be used to solve *L* in polynomial time.^{ [1] }^{ [2] } As a consequence, finding a polynomial time algorithm to solve any NP-hard problem would give polynomial time algorithms for all the problems in NP. As it is suspected that P NP, it is unlikely that such an algorithm exists.^{ [3] }

A common misconception is that the *NP* in "NP-hard" stands for "non-polynomial" when in fact it stands for "non-deterministic polynomial acceptable problems".^{ [4] } It is suspected that there are no polynomial-time algorithms for NP-hard problems, but that has not been proven.^{ [5] } Moreover, the class P, in which all problems can be solved in polynomial time, is contained in the NP class.^{ [6] }

A decision problem *H* is NP-hard when for every problem *L* in NP, there is a polynomial-time many-one reduction from *L* to *H*.^{ [1] }^{:80} An equivalent definition is to require that every problem *L* in NP can be solved in polynomial time by an oracle machine with an oracle for *H*.^{ [7] } Informally, an algorithm can be thought of that calls such an oracle machine as a subroutine for solving *H* and solves *L* in polynomial time if the subroutine call takes only one step to compute.

Another definition is to require that there be a polynomial-time reduction from an NP-complete problem *G* to *H*.^{ [1] }^{:91} As any problem *L* in NP reduces in polynomial time to *G*, *L* reduces in turn to *H* in polynomial time so this new definition implies the previous one. Awkwardly, it does not restrict the class NP-hard to decision problems, and it also includes search problems or optimization problems.

If P ≠ NP, then NP-hard problems cannot be solved in polynomial time.

Some NP-hard optimization problems can be polynomial-time approximated up to some constant approximation ratio (in particular, those in APX) or even up to any approximation ratio (those in PTAS or FPTAS).

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. Another example of an NP-hard problem is the optimization problem of finding the least-cost cyclic route through all nodes of a weighted graph. This is commonly known as the traveling salesman problem.^{ [8] }

There are decision problems that are *NP-hard* but not *NP-complete* such as the halting problem. That is the problem which asks "given a program and its input, will it run forever?" That is a *yes*/*no* question and so is a decision problem. It is easy to prove that the halting problem is NP-hard but not NP-complete. For example, the Boolean satisfiability problem can be reduced to the halting problem by transforming it to the description of a Turing machine that tries all truth value assignments and when it finds one that satisfies the formula it halts and otherwise it goes into an infinite loop. It is also easy to see that the halting problem is not in *NP* since all problems in NP are decidable in a finite number of operations, but the halting problem, in general, is undecidable. There are also NP-hard problems that are neither *NP-complete* nor *Undecidable*. For instance, the language of true quantified Boolean formulas is decidable in polynomial space, but not in non-deterministic polynomial time (unless NP = PSPACE).^{ [9] }

NP-hard problems do not have to be elements of the complexity class NP. As NP plays a central role in computational complexity, it is used as the basis of several classes:

- NP
- Class of computational decision problems for which a given
*yes*-solution can be verified as a solution in polynomial time by a deterministic Turing machine (or*solvable*by a*non-deterministic*Turing machine in polynomial time). - NP-hard
- Class of problems which are at least as hard as the hardest problems in NP. Problems that are NP-hard do not have to be elements of NP; indeed, they may not even be decidable.
- NP-complete
- Class of decision problems which contains the hardest problems in NP. Each NP-complete problem has to be in NP.
- NP-easy
- At most as hard as NP, but not necessarily in NP.
- NP-equivalent
- Decision problems that are both NP-hard and NP-easy, but not necessarily in NP.
- NP-intermediate
- If P and NP are different, then there exist decision problems in the region of NP that fall between P and the NP-complete problems. (If P and NP are the same class, then NP-intermediate problems do not exist because in this case every NP-complete problem would fall in P, and by definition, every problem in NP can be reduced to an NP-complete problem.)

NP-hard problems are often tackled with rules-based languages in areas including:

- Approximate computing
- Configuration
- Cryptography
- Data mining
- Decision support
- Phylogenetics
- Planning
- Process monitoring and control
- Rosters or schedules
- Routing/vehicle routing
- Scheduling

In computational complexity theory, **bounded-error probabilistic polynomial time** (**BPP**) is the class of decision problems solvable by a probabilistic Turing machine in polynomial time with an error probability bounded away from 1/3 for all instances. **BPP** is one of the largest *practical* classes of problems, meaning most problems of interest in **BPP** have efficient probabilistic algorithms that can be run quickly on real modern machines. **BPP** also contains **P**, the class of problems solvable in polynomial time with a deterministic machine, since a deterministic machine is a special case of a probabilistic machine.

The **P versus NP problem** is a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly verified can also be solved quickly.

**Computational complexity theory** focuses on classifying computational problems according to their inherent difficulty, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm.

In computational complexity theory, **NP** is a complexity class used to classify decision problems. NP is the set of decision problems for which the problem instances, where the answer is "yes", have proofs verifiable in polynomial time by a deterministic Turing machine.

In complexity theory and computability theory, an **oracle machine** is an abstract machine used to study decision problems. It can be visualized as a Turing machine with a black box, called an **oracle**, which is able to solve certain decision problems in a single operation. The problem can be of any complexity class. Even undecidable problems, such as the halting problem, can be used.

In computational complexity theory, **PSPACE** is the set of all decision problems that can be solved by a Turing machine using a polynomial amount of space.

In computational complexity theory, a decision problem is **PSPACE-complete** if it can be solved using an amount of memory that is polynomial in the input length and if every other problem that can be solved in polynomial space can be transformed to it in polynomial time. The problems that are PSPACE-complete can be thought of as the hardest problems in **PSPACE**, because a solution to any one such problem could easily be used to solve any other problem in **PSPACE**.

In complexity theory, the complexity class **NP-easy** is the set of function problems that are solvable in polynomial time by a deterministic Turing machine with an oracle for some decision problem in NP.

In computational complexity theory, the complexity class **EXPTIME** is the set of all decision problems that are solvable by a deterministic Turing machine in exponential time, i.e., in O(2^{p}) time, where *p*(*n*) is a polynomial function of *n*. EXPTIME is one class in an exponential hierarchy of complexity classes with increasingly more complex oracles or quantifier alternations. For example, the class 2-EXPTIME is defined similarly to EXPTIME but with a doubly exponential time bound . This can be generalized to higher and higher time bounds. EXPTIME can also be reformulated as the space class APSPACE, the set of all problems that can be solved by an alternating Turing machine in polynomial space.

In computational complexity theory, **EXPSPACE** is the set of all decision problems solvable by a deterministic Turing machine in exponential space, i.e., in space, where is a polynomial function of . Some authors restrict to be a linear function, but most authors instead call the resulting class ESPACE. If we use a nondeterministic machine instead, we get the class NEXPSPACE, which is equal to EXPSPACE by Savitch's theorem.

In computational complexity theory, a **polynomial-time reduction** is a method for solving one problem using another. One shows that if a hypothetical subroutine solving the second problem exists, then the first problem can be solved by transforming or reducing it to inputs for the second problem and calling the subroutine one or more times. If both the time required to transform the first problem to the second, and the number of times the subroutine is called is polynomial, then the first problem is polynomial-time reducible to the second.

In computational complexity theory, a **complexity class** is a set of problems of related resource-based complexity. The two most common resources considered are time and memory.

In computational complexity theory, **P**, also known as **PTIME** or **DTIME**(*n*^{O }), is a fundamental complexity class. It contains all decision problems that can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time.

In complexity theory, **PP** is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of less than 1/2 for all instances. The abbreviation **PP** refers to probabilistic polynomial time. The complexity class was defined by Gill in 1977.

In computational complexity theory, the **Cook–Levin theorem**, also known as **Cook's theorem**, states that the Boolean satisfiability problem is NP-complete. That is, it is in NP, and any problem in NP can be reduced in polynomial time by a deterministic Turing machine to the Boolean satisfiability problem.

In computational complexity theory, a **function problem** is a computational problem where a single output is expected for every input, but the output is more complex than that of a decision problem. For function problems, the output is not simply 'yes' or 'no'.

In computational complexity theory, the complexity class **NEXPTIME** is the set of decision problems that can be solved by a non-deterministic Turing machine using time .

In computability theory and computational complexity theory, a **reduction** is an algorithm for transforming one problem into another problem. A sufficiently efficient reduction from one problem to another may be used to show that the second problem is at least as difficult as the first.

In computational complexity theory, a problem is **NP-complete** when it can be solved by a restricted class of brute force search algorithms and it can be used to simulate any other problem with a similar algorithm. More precisely, each input to the problem should be associated with a set of solutions of polynomial length, whose validity can be tested quickly, 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, and a problem is NP-complete if it is both in NP and NP-hard. 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. The set of NP-complete problems is often denoted by **NP-C** or **NPC**.

**Quantum complexity theory** is the subfield of computational complexity theory that deals with complexity classes defined using quantum computers, a computational model based on quantum mechanics. It studies the hardness of computational problems in relation to these complexity classes, as well as the relationship between quantum complexity classes and classical complexity classes.

- 1 2 3 Leeuwen, Jan van, ed. (1998).
*Handbook of Theoretical Computer Science*. Vol. A, Algorithms and complexity. Amsterdam: Elsevier. ISBN 0262720140. OCLC 247934368. - ↑ Knuth, Donald (1974). "Postscript about NP-hard problems".
*ACM SIGACT News*.**6**(2): 15–16. doi:10.1145/1008304.1008305. - ↑ Daniel Pierre Bovet; Pierluigi Crescenzi (1994).
*Introduction to the Theory of Complexity*. Prentice Hall. p. 69. ISBN 0-13-915380-2. - ↑ "P and NP".
*www.cs.uky.edu*. Archived from the original on 2016-09-16. Retrieved 2016-09-25. - ↑ "Shtetl-Optimized » Blog Archive » The Scientific Case for P≠NP".
*www.scottaaronson.com*. Retrieved 2016-09-25. - ↑ "PHYS771 Lecture 6: P, NP, and Friends".
*www.scottaaronson.com*. Retrieved 2016-09-25. - ↑ V. J. Rayward-Smith (1986).
*A First Course in Computability*. Blackwell. p. 159. ISBN 0-632-01307-9. - ↑ Lawler, E. L.; Lenstra, J. K.; Rinnooy Kan, A. H. G.; Shmoys, D. B. (1985),
*The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization*, John Wiley & Sons, ISBN 0-471-90413-9 . - ↑ More precisely, this language is PSPACE-complete; see, for example, Wegener, Ingo (2005),
*Complexity Theory: Exploring the Limits of Efficient Algorithms*, Springer, p. 189, ISBN 9783540210450 .

- Michael R. Garey and David S. Johnson (1979).
*Computers and Intractability: A Guide to the Theory of NP-Completeness*. W.H. Freeman. ISBN 0-7167-1045-5.

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