I was given a problem in which you are supposed to write a python code that distributes a number of different weights among 4 boxes.

Logically we can't expect a perfect distribution as in case we are given weights like 10, 65, 30, 40, 50 and 60 kilograms, there is no way of grouping those numbers without making one box heavier than another. But we can aim for the most homogenous distribution. ((60),(40,30),(65),(50,10))

I can't even think of an algorithm to complete this task let alone turn it into python code. Any ideas about the subject would be appreciated.

CodePudding user response：

The problem you're describing is similar to the "fair teams" problem, so I'd suggest looking there first.

Because a simple greedy algorithm where weights are added to the lightest box won't work, the most straightforward solution would be a brute force recursive backtracking algorithm that keeps track of the best solution it has found while iterating over all possible combinations.

CodePudding user response：

As stated in @j_random_hacker's response, this is not going to be something easily done. My best idea right now is to find some baseline. I describe a baseline as an object with the largest value since it cannot be subdivided. Using that you can start trying to match the rest of the data to that value which would only take about three iterations to do. The first and second would create a list of every possible combination and then the third can go over that list and compare the different options by taking the average of each group and storing the closest average value to your baseline.

Using your example, 65 is the baseline and since you cannot subdivide it you know that has to be the minimum bound on your data grouping so you would try to match all of the rest of the values to that. It wont be great, but it does give you something to start with.

CodePudding user response：

As j_random_hacker notes, the partition problem is NP-complete. This problem is also NP-complete by a reduction from the 4-partition problem (the article also contains a link to a paper by Garey and Johnson that proves that 4-partition itself is NP-complete).

In particular, given a list to 4-partition, you could feed that list as an input to a function that solves your box distribution problem. If each box had the same weight in it, a 4-partition would exist, otherwise not.

Your best bet would be to create an exponential time algorithm that uses backtracking to iterate over the 4^n possible assignments. Because unless P = NP (highly unlikely), no polynomial time algorithm exists for this problem.