Why knapsack problem is pseudo-polynomial?

5 Answers. The running time is O(NW) for an unbounded knapsack problem with N items and knapsack of size W. W is not polynomial in the length of the input though, which is what makes it pseudo-polynomial.

Can knapsack be solved in polynomial time?

A polynomial-time algorithm is presented and analyzed However, ~t remains an open problem that for any fixed n > 2, the knapsack problem with n variables can be solved in polynomial time.

What is pseudo-polynomial runtime?

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 the input (the number of bits required to represent it), which is the case for …

Is knapsack a polynomial time problem?

Computational complexity The decision problem form of the knapsack problem (Can a value of at least V be achieved without exceeding the weight W?) is NP-complete, thus there is no known algorithm both correct and fast (polynomial-time) in all cases. There is a pseudo-polynomial time algorithm using dynamic programming.

Is dynamic programming pseudo polynomial?

Pseudo-polynomial and NP-Completeness For example, Dynamic Programming Solutions of 0-1 Knapsack, Subset-Sum and Partition problems are Pseudo-Polynomial.

What is knapsack problem with example?

0/1 Knapsack Problem: Items are indivisible; you either take an item or not. Some special instances can be solved with dynamic programming. b. Fractional knapsack problem: Items are divisible; you can take any fraction of an item….Welcome back.

i	Pi	Wi
3	5	4
4	6	5

Is knapsack problem in P?

We call such algorithms pseudo-polynomial time algorithms. Hence, we see that Knapsack is not NP-complete if the given input is unary (assuming P = NP), but NP-complete when the given input is binary. Such problems are called weakly NP- complete.

What is greedy knapsack problem?

The basic idea of the greedy approach is to calculate the ratio value/weight for each item and sort the item on basis of this ratio. Then take the item with the highest ratio and add them until we can’t add the next item as a whole and at the end add the next item as much as we can.

Is Fibonacci pseudo-polynomial?

2 Answers. “Fib(n) is pseudo-polynomial” means in this context that computing Fib is bounded by a polynomial of its argument, n, but isn’t bounded by a polynomial function of the size of the argument, log(n).

Is pseudo-polynomial NP?

When the complexity of an algorithm is polynomial in the size of the data, but not the length of the input, we refer to it as a pseudo-polynomial algorithm. A NP-complete problem solvable by a pseudo-polynomial algorithm is called ordinary NP-complete.

What is considered polynomial time?

(definition) Definition: When the execution time of a computation, m(n), is no more than a polynomial function of the problem size, n. More formally m(n) = O(nk) where k is a constant.

What is knapsack DP?

So the 0-1 Knapsack problem has both properties (see this and this) of a dynamic programming problem. The state DP[i][j] will denote maximum value of ‘j-weight’ considering all values from ‘1 to ith’. So if we consider ‘wi’ (weight in ‘ith’ row) we can fill it in all columns which have ‘weight values > wi’.

Is the knapsack problem a pseudo polynomial problem?

The knapsack problem is an old and popular optimization problem. In this tutorial, we’ll look at different variants of the Knapsack problem and discuss the 0-1 variant in detail. Furthermore, we’ll discuss why it is an NP-Complete problem and present a dynamic programming approach to solve it in pseudo-polynomial time.

Are there any NP complete problems that are pseudo polynomial?

Some NP Complete problems have Pseudo Polynomial time solutions. For example, Dynamic Programming Solutions of 0-1 Knapsack, Subset-Sum and Partition problems are Pseudo-Polynomial. NP complete problems that can be solved using a pseudo-polynomial time algorithms are called weakly NP-complete.

Is the 0-1 knapsack problem NP complete?

The decision version of the 0-1 knapsack problem is an NP-Complete problem. Let’s see why. Given weights and values of items, and , respectively, can a subset of items be picked that satisfy the following constraints: A ‘Yes’ or ‘No’ solution to the above decision problem is NP-Complete.

Which is the best definition of the knapsack problem?

Given a knapsack with a weight limit of , a collection of items with values and weights , the knapsack problem is defined as the optimization problem: Now, the question is, what is the maximum value of the items that can be added to the knapsack such that the weight does not exceed the weight limit ? 3. Variants of Knapsack Problem

Why knapsack problem is pseudo-polynomial? 5 Answers. The running time is O(NW) for an unbounded knapsack problem with N items and knapsack of size W. W is not polynomial in the length of the input though, which is what makes it pseudo-polynomial. Can knapsack be solved in polynomial time? A polynomial-time algorithm is presented and…