Weakly NP-complete
Encyclopedia
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 involved (provided these are given as integers), rather than the base-two logarithms of their magnitudes. Such algorithms are technically exponential functions of their input size and are therefore not considered polynomial.
For example, the NP-hard knapsack problem
can be solved by a dynamic programming
algorithm requiring a number of steps polynomial in the size of the knapsack and the number of items (assuming that all data are scaled to be integers). This algorithm is exponential time since the input sizes of the objects and knapsack are logarithmic in their magnitudes. However, as Garey and Johnson (1979) observed, “A pseudo-polynomial-time
algorithm … will display 'exponential behavior' only when confronted with instances containing 'exponentially large' numbers, [which] might be rare for the application we are interested in. If so, this type of algorithm might serve our purposes almost as well as a polynomial time algorithm.”
The related term strongly NP-complete
(or unary NP-complete) refers to those problems that remain NP-complete even if the data are encoded in unary
(that is, if the data are “small” relative to the overall input size).
Computational Complexity
Computational Complexity may refer to:*Computational complexity theory*Computational Complexity...
, an NP-complete
NP-complete
In computational complexity theory, the complexity class NP-complete is a class of decision problems. A decision problem L is NP-complete if it is in the set of NP problems so that any given solution to the decision problem can be verified in polynomial time, and also in the set of NP-hard...
(or NP-hard
NP-hard
NP-hard , in computational complexity theory, is a class of problems that are, informally, "at least as hard as the hardest problems in NP". A problem H is NP-hard if and only if there is an NP-complete problem L that is polynomial time Turing-reducible to H...
) 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 involved (provided these are given as integers), rather than the base-two logarithms of their magnitudes. Such algorithms are technically exponential functions of their input size and are therefore not considered polynomial.
For example, the NP-hard knapsack problem
Knapsack problem
The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the count of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as...
can be solved by a dynamic programming
Dynamic programming
In mathematics and computer science, dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It is applicable to problems exhibiting the properties of overlapping subproblems which are only slightly smaller and optimal substructure...
algorithm requiring a number of steps polynomial in the size of the knapsack and the number of items (assuming that all data are scaled to be integers). This algorithm is exponential time since the input sizes of the objects and knapsack are logarithmic in their magnitudes. However, as Garey and Johnson (1979) observed, “A pseudo-polynomial-time
Pseudo-polynomial time
In computational complexity theory, a numeric algorithm runs in pseudo-polynomial time if its running time is polynomial in the numeric value of the input ....
algorithm … will display 'exponential behavior' only when confronted with instances containing 'exponentially large' numbers, [which] might be rare for the application we are interested in. If so, this type of algorithm might serve our purposes almost as well as a polynomial time algorithm.”
The related term strongly NP-complete
Strongly NP-complete
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...
(or unary NP-complete) refers to those problems that remain NP-complete even if the data are encoded in unary
Unary numeral system
The unary numeral system is the bijective base-1 numeral system. It is the simplest numeral system to represent natural numbers: in order to represent a number N, an arbitrarily chosen symbol representing 1 is repeated N times. For example, using the symbol | , the number 6 is represented as ||||||...
(that is, if the data are “small” relative to the overall input size).