NP-hardness

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Euler diagram for P, NP, NP-complete, and NP-hard set of problems. The left side is valid under the assumption that P[?]NP, while the right side is valid under the assumption that P=NP (except that the empty language and its complement are never NP-complete) P np np-complete np-hard.svg
Euler diagram for P, NP, NP-complete, and NP-hard set of problems. The left side is valid under the assumption that P≠NP, while the right side is valid under the assumption that P=NP (except that the empty language and its complement are never NP-complete)

In computational complexity theory, a computational problem H is called NP-hard if, for every problem L which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from L 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 a single NP-hard problem would give polynomial time algorithms for all the problems in the complexity class NP. As it is suspected that P≠NP, it is unlikely that such an algorithm exists. [3] It is suspected that there are no polynomial-time algorithms for NP-hard problems, but that has not been proven. [4] A simple example of an NP-hard problem is the subset sum problem.

Contents

Informally, if H is NP-hard, then it is at least as difficult to solve as the problems in NP. However, the opposite direction is not true: some problems are undecidable, and therefore even more difficult to solve than all problems in NP, but they are provably not NP-hard (unless P=NP). [5]

Definition

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 

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. It does not restrict the class NP-hard to decision problems, and it also includes search problems or optimization problems.

Consequences

If P ≠ NP, then NP-hard problems could not 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). There are many classes of approximability, each one enabling approximation up to a different level. [6]

Examples

All NP-complete problems are also NP-hard (see List of NP-complete problems). For example, the optimization problem of finding the least-cost cyclic route through all nodes of a weighted graph—commonly known as the travelling salesman problem —is NP-hard. [7] The subset sum problem is another example: 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.

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). [8]

NP-naming convention

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 any 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.)

Application areas

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

See also

Related Research Articles

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In computational complexity theory, co-NP is a complexity class. A decision problem X is a member of co-NP if and only if its complement X is in the complexity class NP. The class can be defined as follows: a decision problem is in co-NP if and only if for every no-instance we have a polynomial-length "certificate" and there is a polynomial-time algorithm that can be used to verify any purported certificate.

In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, 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.

<span class="mw-page-title-main">Decision problem</span> Yes/no problem in computer science

In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm whether a given natural number is prime. Another is the problem "given two numbers x and y, does x evenly divide y?". The answer is either 'yes' or 'no' depending upon the values of x and y. A method for solving a decision problem, given in the form of an algorithm, is called a decision procedure for that problem. A decision procedure for the decision problem "given two numbers x and y, does x evenly divide y?" would give the steps for determining whether x evenly divides y. One such algorithm is long division. If the remainder is zero the answer is 'yes', otherwise it is 'no'. A decision problem which can be solved by an algorithm is called decidable.

<span class="mw-page-title-main">NP (complexity)</span> Complexity class used to classify decision problems

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, or alternatively the set of problems that can be solved in polynomial time by a nondeterministic 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 problems in a single operation. The problem can be of any complexity class. Even undecidable problems, such as the halting problem, can be used.

<span class="mw-page-title-main">PSPACE</span> Set of decision problems

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.

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In computational complexity theory, the complexity class EXPTIME (sometimes called EXP or DEXPTIME) is the set of all decision problems that are solvable by a deterministic Turing machine in exponential time, i.e., in O(2p(n)) time, where p(n) is a polynomial function of n.

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 computability theory and computational complexity theory, a many-one reduction is a reduction which converts instances of one decision problem to another decision problem using an effective function. The reduced instance is in the language if and only if the initial instance is in its language . Thus if we can decide whether instances are in the language , we can decide whether instances are in its language by applying the reduction and solving for . Thus, reductions can be used to measure the relative computational difficulty of two problems. It is said that reduces to if, in layman's terms is at least as hard to solve as . This means that any algorithm that solves can also be used as part of a program that solves .

<span class="mw-page-title-main">Complexity class</span> Set of problems in computational complexity theory

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In computational complexity theory, P, also known as PTIME or DTIME(nO(1)), 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.

<span class="mw-page-title-main">PP (complexity)</span> Class of problems in computer science

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.

<span class="mw-page-title-main">Reduction (complexity)</span>

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, P/poly is a complexity class representing problems that can be solved by small circuits. More precisely, it is the set of formal languages that have polynomial-size circuit families. It can also be defined equivalently in terms of Turing machines with advice, extra information supplied to the Turing machine along with its input, that may depend on the input length but not on the input itself. In this formulation, P/poly is the class of decision problems that can be solved by a polynomial-time Turing machine with advice strings of length polynomial in the input size. These two different definitions make P/poly central to circuit complexity and non-uniform complexity.

<span class="mw-page-title-main">NP-completeness</span> Complexity class

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

  1. It is a decision problem, meaning that for any input to the problem, the output is either "yes" or "no".
  2. When the answer is "yes", this can be demonstrated through the existence of a short solution.
  3. The correctness of each solution can be verified quickly and a brute-force search algorithm can find a solution by trying all possible solutions.
  4. The problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified.

References

  1. 1 2 3 Leeuwen, Jan van, ed. (1998). Handbook of Theoretical Computer Science. Vol. A, Algorithms and complexity. Amsterdam: Elsevier. ISBN   0262720140. OCLC   247934368.
  2. ↑ Knuth, Donald (1974). "Postscript about NP-hard problems". ACM SIGACT News. 6 (2): 15–16. doi:10.1145/1008304.1008305. S2CID   46480926.
  3. ↑ Daniel Pierre Bovet; Pierluigi Crescenzi (1994). Introduction to the Theory of Complexity. Prentice Hall. p. 69. ISBN   0-13-915380-2.
  4. ↑ "Shtetl-Optimized » Blog Archive » The Scientific Case for P≠NP". www.scottaaronson.com. Retrieved 2016-09-25.
  5. ↑ "Is undecidable(complement of R) a subset of NP-hard?". Computer Science Stack Exchange. Retrieved 2024-02-09.
  6. ↑ Escoffier, B.; Paschos, B.Th. (2010). "A survey on the structure of approximation classes". Computer Science Review. 4 (1): 19–40.
  7. ↑ 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 .
  8. ↑ 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 .
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