In computational complexity theory, a numeric algorithm runs in pseudo-polynomial time if its running time is bounded from above by a polynomial function of the two variables: the numeric value of the input (the largest integer present in the input) and the length of the input (the number of bits required to represent it).Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, 1979.
In general, using a positional number system, the numeric value of the input is exponential in the input length, which is why a pseudo-polynomial time algorithm does not necessarily run in polynomial time with respect to the input length. The distinction between the value of a number and its length is due to positional encoding; if numeric inputs are instead encoded in unary then the length and value are the same.
An NP-complete problem with known pseudo-polynomial time algorithms is called weakly NP-complete. An NP-complete problem is called strongly NP-complete if it is proven that it cannot be solved by a pseudo-polynomial time algorithm unless . The strong/weak kinds of NP-hardness are defined analogously. Strong NP-completeness of a problem can be established by a pseudo-polynomial transformation from a known strongly NP-complete problem.
Examples
Primality testing Consider solving the problem of testing whether a number n is prime by naively checking whether a number in divides evenly. This approach can take up to divisions, which is sub-linear in the value of n but exponential in the length of n (which is about ). For example, a number n slightly less than would require up to approximately 100,000 divisions, even though the length of n is only 11 digits. Moreover one can easily write down an input (say, a 300-digit number) for which this algorithm is impractical. Since computational complexity measures difficulty with respect to the length of the (encoded) input, this naive algorithm is actually exponential. It is, however, pseudo-polynomial time.
Contrast this algorithm with a true polynomial numeric algorithm—say, the straightforward algorithm for addition: Adding two 9-digit numbers takes around 9 simple steps, and in general the algorithm is truly linear in the length of the input. Compared with the actual numbers being added (in the billions), the algorithm could be called "pseudo-logarithmic time", though such a term is not standard. Thus, adding 300-digit numbers is not impractical. Similarly, long division is quadratic: an m-digit number can be divided by a n-digit number in steps (see Big O notation.)
In the case of primality, it turns out there is a different algorithm for testing whether n is prime (discovered in 2002) that runs in time .
Knapsack problem In the knapsack problem, we are given items with weight and value , along with a maximum weight capacity of a knapsack . The goal is to solve the following optimization problem; informally, what's the best way to fit the items into the knapsack to maximize value? maximize subject to and . Solving this problem is NP-hard, so a polynomial time algorithm is impossible unless . However, an time algorithm is possible using dynamic programming; since the number only needs bits to describe, this algorithm runs in pseudo-polynomial time.
See also Strongly NP-complete Quasi-polynomial time Pseudo-polynomial transformation
References