In computational complexity, strong NP-completeness is a property of computational problems that is a special case of

NP-complete In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying ...

ness. A general computational problem may have numerical parameters. For example, the input to the bin packing problem is a list of objects of specific sizes and a size for the bins that must contain the objects—these object sizes and bin size are numerical parameters. A problem is said to be strongly

(NP-complete in the strong sense), if it remains NP-complete even when all of its numerical parameters are bounded by a polynomial in the length of the input. A problem is said to be strongly NP-hard if a strongly NP-complete problem has a polynomial reduction to it; in combinatorial optimization, particularly, the phrase "strongly NP-hard" is reserved for problems that are not known to have a polynomial reduction to another strongly NP-complete problem. Normally numerical parameters to a problem are given in positional notation, so a problem of input size ''n'' might contain parameters whose size is

exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Expo ...

in ''n''. If we redefine the problem to have the parameters given in unary notation, then the parameters must be bounded by the input size. Thus strong NP-completeness or NP-hardness may also be defined as the NP-completeness or NP-hardness of this unary version of the problem. For example, bin packing is strongly NP-complete while the

0-1 Knapsack problem The knapsack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit an ...

is only

weakly NP-complete In computational complexity, an NP-complete (or NP-hard) problem is weakly NP-complete (or weakly NP-hard) if there is an algorithm for the problem whose running time is polynomial in the dimension of the problem and the magnitudes of the data inv ...

. Thus the version of bin packing where the object and bin sizes are integers bounded by a polynomial remains NP-complete, while the corresponding version of the Knapsack problem can be solved in

pseudo-polynomial time In computational complexity theory, a numeric algorithm runs in pseudo-polynomial time if its running time is a polynomial in the ''numeric value'' of the input (the largest integer present in the input)—but not necessarily in the ''length'' of t ...

dynamic programming Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. ...

. From a theoretical perspective any strongly NP-hard optimization problem with a polynomially bounded objective function cannot have a

fully polynomial-time approximation scheme A fully polynomial-time approximation scheme (FPTAS) is an algorithm for finding approximate solutions to function problems, especially optimization problems. An FPTAS takes as input an instance of the problem and a parameter ε > 0. It r ...

(or

FPTAS A fully polynomial-time approximation scheme (FPTAS) is an algorithm for finding approximate solutions to function problems, especially optimization problems. An FPTAS takes as input an instance of the problem and a parameter ε > 0. It r ...

) unless P = NP. However, the converse fails: e.g. if P does not equal NP, knapsack with two constraints is not strongly NP-hard, but has no FPTAS even when the optimal objective is polynomially bounded. Some strongly NP-complete problems may still be easy to solve ''on average'', but it's more likely that difficult instances will be encountered in practice.

References

{{Reflist Strongly NP-complete problems Complexity classes Computational complexity theory