## Knapsack

### August 23, 2011

We look today at a famous problem known as the knapsack problem. The task is to fill a knapsack that can carry objects of a known total weight—the capacity—with objects of a given set—each object having a given weight and value—in such a way as to maximize the sum of the values of the objects. Since any individual object is subject to a binary decision to either take it or leave it, this variant of the knapsack problem is known as the 0/1 knapsack.

The usual algorithm uses dynamic programming on the recurrence expression *V*[*i*,*w*] = max(*V*[*i*−1,*w*], *v _{i}* +

*V*[

*i*−1,

*w*−

*w*]), where

_{i}*V*[

*i*,

*w*] is the maximum value of a knapsack of capacity

*w*using the first

*i*objects (which must be arranged by increasing weight),

*v*is the value of the

_{i}*i*th object, and

*w*is the weight of the

_{i}*i*th object. The program builds up the

*V*matrix, starting from

*V*[0,

*w*] = 0, for all weights 0 ≤

*w*≤

*W*and all sets of objects 1 ≤

*i*≤

*n*, where

*n*is the number of objects available. The answer is

*V*[

*n*,

*W*].

Your task is to write a function that implements the knapsack algorithm. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

Python 3.

‘items’ is a list of objects that have ‘name’, ‘value’, and ‘weight’ attributes. For testing, I used the Item() namedtuple above.

Example,

What is W? Why do you start with V[0,w] and not, say, V[i,w], which would make more sense since you wrote “1 ≤ i ≤ n”

wis the weight of the_{i}ith object.V[0,w] is the base of the recurrence expression, and hence must be calculated first.I was asking about the capital W, which is defined nowhere. I guess it’s the maximum weight one can bear… Also, it should be “all objects 1≤i≤n” rather than “all sets of objects 1≤i≤n” shouldn’t it?

Anyway, I have a solution, but it’s too ugly for posting :)

W is the capacity of the knapsack.

(define values

#(2 4 5 1 3))

(define weights

#(3 2 2 3 4))

(define (knapsack w i)

(cond

((or (= w 0)

(= i 0)) 0)

((> (vector-ref weights (sub1 i)) w)

(knapsack w (sub1 i)))

(else

(max (knapsack w (sub1 i))

(+ (vector-ref values (sub1 i))

(knapsack (- w (vector-ref weights (sub1 i)))

(- i 1)))))))

Sorry for poor formating

Why must the objects be sorted by weight? Your examples don’t obey this constraint, and your program doesn’t sort the objects. Anyway, I didn’t bother with the matrix and just used recursion and memoization (like in the Word Breaks problem):

[…] reduces the multiplication of the factors to addition of their logarithms, which means that the knapsack algorithm can be used to find the greatest sum less than the logarithm of the square […]