The Knapsack Problem
CMPSC 465 – Kleinberg/Tardos 6.4
I. The Problem
Here’s the scenario for the knapsack problem:
• We have n potential items we could pick up and a knapsack to put them in
• Each item i…
o has a weight of wi kg
o has a value of vi s.t. vi > 0
• The knapsack has a capacity of W kg
• Our goal is to fill the knapsack while maximizing the total value
Example Instance: Let’s say we have a knapsack of capacity 11 kg. Here’s a set of weights and values:
II. Diversion: Greedy
Problem: Take a few minutes and work with a partner to propose a greedy algorithm to fill your knapsack. Why is it greedy?
Try it out on the example instance above.
pick item with max ratio of value/weight
{5, 2, 1} is output but gets value 35. but {3, 4} gets 40. greedy not optimal
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Prepared by D. Hogan referencing Algorithm Design by Kleinberg/Tardos and slides by K. Wayne for PSU CMPSC 465
III. Building a Solution via Dynamic Programming
Recall what we did with the weighted interval scheduling problem:
• We called an optimal solution for the first i itervals Oi
• We defined OPT(i) to be the value of an optimal solution using the first i intervals
• We figured out that recursively computing subproblems resulted in solving the same subproblem many times, so we
instead memoized and iteratively built up a table of the values of OPT(i).
So, let’s see how far we can get building up an OPT(i) for knapsack.
Let’s define OPT(i) as…
There are two cases to consider:
•
Item i is not selected in an optimal solution
•
Item i is selected in an optimal solution
Observations:
So, let’s tweak our definition of OPT(i) and make it OPT(i, w):
Or two cases now change as follows:
•
Item i is not selected in an optimal solution with weight limit w
•
Item i is selected in an optimal solution
Page 2 of 2
Prepared by D. Hogan referencing Algorithm Design by Kleinberg/Tardos and slides by K. Wayne for PSU CMPSC 465
We can summarize via the following function:
Now, let’s turn that function into an iterative algorithm:
Page 3 of 3
Prepared by D. Hogan referencing Algorithm Design by Kleinberg/Tardos and slides by K. Wayne for PSU CMPSC 465
IV. Example
Back to this example instance: Our knapsack has capacity 11 kg. Here’s a set of weights and values:
Use the algorithm we just developed to solve this instance of the knapsack problem.
weights
n
items
selected
from
0
1
2
3
4
5
6
7
8
9
10
11
0
1
2
3
4
5
V. Analysis
Question: What is the runtime of this algorithm?
Theta(nW)
NP-complete
Homework:
1. Describe an algorithm to recover the optimal subset from the table M generated by our algorithm and the algorithm’s
inputs.
2. Try the knapsack algorithm on an input with items of weights 3 lb., 2 lb., 1 lb., 4 lb., 5 lb. and values $25, $20, $15,
$40, and $50, respectively. The knapsack capacity is 6 lb. (Levitin Exercise 8.4.1a)
I want to prove the correctness of that algorithm somewhere – maybe recitation?
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Prepared by D. Hogan referencing Algorithm Design by Kleinberg/Tardos and slides by K. Wayne for PSU CMPSC 465