How to understand the knapsack problem is NP-complete?
AlgorithmComplexity TheoryAlgorithm Problem Overview
We know that the knapsack problem can be solved in O(nW) complexity by dynamic programming. But we say this is a NP-complete problem. I feel it is hard to understand here.
(n is the number of items. W is the maximum volume.)
Algorithm Solutions
Solution 1 - Algorithm
O(n*W) looks like a polynomial time, but it is not, it is pseudo-polynomial.
Time complexity measures the time that an algorithm takes as a function of the length in bits of its input. The dynamic programming solution is indeed linear in the value of W, but exponential in the length of W — and that's what matters!
More precisely, the time complexity of the dynamic solution for the knapsack problem is basically given by a nested loop:
// here goes other stuff we don't care about
for (i = 1 to n)
for (j = 0 to W)
// here goes other stuff
Thus, the time complexity is clearly O(n*W).
What does it mean to increase linearly the size of the input of the algorithm? It means using progressively longer item arrays (so n, n+1, n+2, ...) and progressively longer W (so, if W is x bits long, after one step we use x+1 bits, then x+2 bits, ...). But the value of W grows exponentially with x, thus the algorithm is not really polynomial, it's exponential (but it looks like it is polynomial, hence the name: "pseudo-polynomial").
Further References
Solution 2 - Algorithm
In knapsack 0/1 problem, we need 2 inputs(1 array & 1 integer) to solve this problem:
- a array of n items: [n1, n2, n3, ... ], each item with its value index and weight index.
- integer W as maximum acceptable weight
Let's assume n=10 and W=8:
- n = [n1, n2, n3, ... , n10]
- W = 1000 in binary term (4-bit long)
so the time complexity T(n) = O(nW) = O(10*8) = O(80)
If you double the size of n:
n = [n1, n2, n3, ... , n10] -> n = [n1, n2, n3, ... , n20]
so time complexity T(n) = O(nW) = O(20*8) = O(160)
but as you double the size of W, it does not mean W=16, but the length will be twice longer:
W = 1000 -> W = 10000000 in binary term (8-bit long)
so T(n) = O(nW) = O(10*128) = O(1280)
the time needed increases in exponential term, so it's a NPC problem.
Solution 3 - Algorithm
It all depends on which parameters you put inside O(...).
If target weight is limited by number W, then problem has O(n*W) complexity, as you mentioned.
But if weights are way too big and you need algorithm with complexity independent of W, then problem is NP-complete. (O(2^n*n) in most naive implementation).
Solution 4 - Algorithm
The size of input is log(W) bits for the weight (and O(n) for "value" and "weight" arrays).
So, the input size of weight is j = log(W) (and not mere W). So, W = 2ʲ (as binary is used).
The final complexity is O(n * W)
> This O(n * W) can be rewritten as O(n * 2ʲ), which exponential in the size of the input.
So, this solution is not polynomial.
Solution 5 - Algorithm
This is because the knapsack problem has a pseudo-polynomial solution and is thus called weakly NP-Complete (and not strongly NP-Complete).
Solution 6 - Algorithm
You can read this short explanation: The NP-Completeness of Knapsack.
Solution 7 - Algorithm
To understand NP-completeness, you have to learn a bit of complexity theory. However, basically, it's NP-complete because an efficient algorithm for the knapsack problem would also be an efficient algorithm for SAT, TSP and the rest.