# Strong NP-completeness

In computational complexity, **strong NP-completeness** is a property of computational problems that is a special case of NP-completeness. 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 (NP-complete in the strong sense), if it remains so even when all of its numerical parameters are bounded by a polynomial in the length of the input.[1] 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 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 is only weakly NP-complete. 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 by dynamic programming.

While weakly NP-complete problems may admit efficient solutions in practice as long as their inputs are of relatively small magnitude, strongly NP-complete problems do not admit efficient solutions in these cases. From a theoretical perspective any strongly NP-hard optimization problem with a polynomially bounded objective function cannot have a fully polynomial-time approximation scheme (or FPTAS) unless P = NP.[2] [3] 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.[4]

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.

## See also

## References

- Garey, M. R.; Johnson, D. S. (July 1978). "'Strong' NP-Completeness Results: Motivation, Examples, and Implications".
*Journal of the Association for Computing Machinery*. New York, NY: ACM.**25**(3): 499–508. doi:10.1145/322077.322090. ISSN 0004-5411. MR 0478747. - Vazirani, Vijay V. (2003).
*Approximation Algorithms*. Berlin: Springer. pp. 294–295. ISBN 3-540-65367-8. MR 1851303. - Garey, M. R.; Johnson, D. S. (1979). Victor Klee (ed.).
*Computers and Intractability: A Guide to the Theory of NP-Completeness*. A Series of Books in the Mathematical Sciences. San Francisco, Calif.: W. H. Freeman and Co. pp. x+338. ISBN 0-7167-1045-5. MR 0519066. - H. Kellerer and U. Pferschy and D. Pisinger (2004).
*Knapsack Problems*. Springer.CS1 maint: uses authors parameter (link)