# Algorithm to pick elements in one array that sum to a setpoint, and corresponding elements in other array to average to a setpoint?

Note: The input file is an excel file. However, I am only looking for help with the algorithm as I can then code it in VBA.

I need to scan both columns (shown below) to find any number of column 1 values that when summed match the setpoint (E2) and column 2 values that when averaged match the setpoint (E3). One constraint is that the chosen values in column 1 and column 2 must be from the same row (or same 'index', shown in the image highlighted in blue and green).

Explanation of the solution below: What it should do is say 'A2 and A6 sum to 7 (match cell E2), and the corresponding B2 and B6 average to 3' (match cell E3). Therefore, these rows should be selected/highlighted in any appropriate way. (note that technically the first solution would be rows 2 and 3 as they fit the sum/average setpoints also).

Example that does not work as a solution: It can't be A4 and A5 because although they add to 7, B4 and B5 average to 1.5, not the setpoint of 3.

Image of the problem (with a solution highlighted):

A solution for one of the cases (below) works only for the case where there are two rows that will sum/average to the setpoints. It will not work for cases where more than two rows are required. The solution should work for any number of rows required.

Example where three rows would be required (Solution is the first three rows):

Reason for this solution: B6:B8 sums to the setpoint of 8 and C6:C8 average to the setpoint of 3
I am looking for help with an algorithm. If the solution is a large number of rows required (>100), how can I iterate through the combinations efficienctly to find a set of rows that both sums/averages to the setpoints?

It may not be possible to find a solution to both setpoints, in this case finding a solution within an error is also ok

Finding an algorithm is NP-Hard, therefore I am also looking for appropriate simplifications to avoid this. An error term may be an example of one simplification.

In general, no; this problem is NP-hard. Even matching just the sum (without getting into anything about the average) is NP-hard, as it is the subset sum problem. Thus, you should not expect any efficient algorithm for your problem that works for all possible inputs.

I notice that in your examples, the numbers are all small. For that special case, there are reasonably efficient algorithms using dynamic programming: see, e.g., https://en.wikipedia.org/wiki/Subset_sum_problem#Pseudo-polynomial_time_dynamic_programming_solution.

The dynamic programming algorithm in that link describes how to make the sum match, but you can adapt it to make both the sum and average match. In particular, you have an array $$Q[t_1,t_2,i,k]$$, which you will fill in with True if there exists a subset of $$k$$ out of the first $$i$$ rows whose first column sums to $$t_1$$ and whose second column sums to $$t_2$$. You can use dynamic programming to fill in this array (in order of increasing $$i$$) in $$O(N^4M^2)$$ time, where $$N$$ counts the number of rows in your spreadsheet and $$M$$ is the largest number in the spreadsheet. Then, once you've filled in this array, there will be a selection that sums to $$s$$ and averages to $$a$$ if and only if there is some $$k$$ such that $$Q[s,ak,N,k]$$ is True.

I recommend implementing this in a programming language (e.g., VBA, not Excel). Implementing it in Excel might be possible but it sounds painful. See https://cs.stackexchange.com/tags/dynamic-programming/info for our reference material on dynamic programming; if you're not familiar with that subject, you might want to read a textbook introduction to it.

• Thanks for the comprehensive reply. Are there any simplifications I can make to use a simpler algorithm? For example, above I state that I do not need to exactly hit the setpoint, only get close enough. Also I can order the array/column in ascending/descending order and possibly do something like: 'Take the first N rows such that the sum is 80% or so of the setpoint and then use the remaining rows to reach the sum, while moving towards the average. It does not have to be perfect in my case. – NoviceCoder11 Apr 22 at 17:42
• Also you state that I should not expect an algorithm that always works. Can I again simplify to find an algorithm that mostly works? – NoviceCoder11 Apr 22 at 17:44
• @NoviceCoder11, you might have to experiment a bit. There are any number of strategies you could try. For instance, for the sum, you could sort from highest to lowest, then iterate through them and take each one as long as it won't cause the running sum to go over the target (or won't go over the target by too much). For the average, you could try taking each one if it won't cause the average so far to vary from the target by too much. I don't know what'll happen; you might need to experiment with a variety of strategies and see if any of them are good enough for you. – D.W. Apr 22 at 23:57
• If all values in the second column are 100, and you want an average of 1 or 99 or 101 or 10000, no algorithm can work. – gnasher729 Apr 23 at 8:33