Knapsack Problem – Dynamic Programming

Dynamic Programming

Dynamic Programming is one very useful technique for solving problems like the Knapsack problem.

The basic idea of Dynamic programming is to break down a problem in recursive subproblems. Each subproblem must be easy to solve, and its solution stored in some data structure to be used again.

In a schematic drawing, it’s something like this:

Dynamic programming is not a general procedure to be applied in any problem. The problem has to allow it. Richard Bellman, the creator of Dynamic Programming, called it  “Principle of optimallity”. It is like saying “this method works in the problems it works, and doesn’t works in the problems it doesn’t works”. But mathematicians love this kind of concept, because they can ask: which problems has the principle of optimality? Is the Knapsack one of them?

Since this method is not general, the programmer has to design a new method for every different problem and also figure out if the problem has the principle of optimality.

But, despite all these observations, it is a useful way of thinking.

The Algorithm

The detailed algorithm can be found in Wikipedia  (https://en.wikipedia.org/wiki/Dynamic_programming). I’ll sketch here the main ideias.

Suppose the problem is the same of previous post.
Max Weight = 14.
List of itens:

1) It is easy to compute the solution with 1 item for a range of weights.

Array of values

Array of picked itens:

2) We increase the number of itens, and compare current solution versus solution with new item.
Complete array with 2 items.
Proceeding this way, we get the full evaluation matrix.
The optimum is to pick itens 1, 3 and 4, and the value of those items is 14,000.
Note that, in order to get the optimum, I had to calculate every solution for 1 to number of itens and 1 to number weigths. Thus, the computational effort to calculate it is proportional to N*maxW.
The problem happens when N and W are large. It needs a lot of computation processing.
Knapsack is a pseudo polynomial time algorithm. Polynomial because it is proportional to N. But pseudo since it is not quite true it is proportional to N – it is also proportional to W, and W can be very very large. One day I’ll post a more detailed way of thinking of this.

Divide and conquer
Other question I had when studying dynamic programming for the first time was: what is the difference of this method compared to divide and conquer methods?
I think this discussion deserves a post apart, but just to give a simple explanation, divide and conquer has a “horizontal” structure, with subproblems independent of each other, while dynamic programming has a “nested” structure, with subproblems dependent of other subproblems.

Computation implementation
An Excel- VBA implementation of dynamic programming applied to the knapsack problem can be found at https://github.com/asgunzi/KnapsackDynamicProgrammingVBA.
It uses the Random case generator of previous post and generates a solution for this case.

Other Applications
Dynamic programming can be applied in several other cases. Cutting stock problems are also a classic application.
One less known application is in the optimization of forest multiproducts. We have a forest of eucalyptus. Each tree has to be cut in several pieces. Each piece has a different diameter (since the tree is larger in the base). A product is a combination of diameter and lenght. Given a demand, how can I minimize the waste and maximize the use of the forest?

In the next posts, I’ll show alternative methods for solving the Knapsack Problem.

Arnaldo Gunzi.

The knapsack problem

Imagine that you are going to a trip, and everything you have just a knapsack to carry on everything.
The maximum weight you can carry is limited, say 14 kg.
There are a list of itens you’re considering to choose from. Each item has a weight and a value.

Which of the itens do I choose, to maximize the value I can carry on my backpack?

The “Human method”

The first method described is the “human method”. How you and me would choose the itens.
I would choose the itens with greatest value for me, one by one, till the maximum weight.

In this example, I would choose itens 1 and 2, whose sum of values is 13.000 and sum of weights is 14. It is the “greedy” method, heuristically simple but can be suboptimal.

In the end, I would take a look on my selection, and try to find if I can replace one of the chosen itens for two itens not choosed with greater value.
In this example, a simple inspection shows that the item 2 is too heavy for its benefits. It is better to replace it for itens 3 and 4.
The final selection contains itens {1, 3, 4}, with total value 14.000 (10.000 + 2000 + 2000) and weight 13 (8 +2+3).
The problem is when the list is very large, with tens or hundreds of itens. It is several timer harder to figure out a good solution.
The human method is very important. It is suboptimal, but it is fast. Humans are good problem solvers. We use intuition and millions of years of understanding of nature, to solve this kind of problem. Human reasoning can be a source for fast approximation algorithms.
Even when the problem is large, humans can solve in a heuristically interesting way. The world is also Pareto, what means that 20% of itens will have 80% of the value. In a intuitive way, I would focus my attention in those 20% and apply the above described method.
In real world application problems, I always respect a lot the people who make the analysis and try to someway incorporate some of theirs thinking in my models – a human centered analytics.

Mathematical formulation

The formulation of the knapsack problem via integer programming is straightforward.
If xi is the decision variable, binary (1 is to carry the item, 0 is to not carry) then
In the next posts I’ll show how to solve it with computational methods, as dynamic programming,  integer programming using solver and evolutive metaheuristics.
Arnaldo Gunzi
A big fan of real world heuristics