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I need a way of finding an exact value made of the sum of variables chosen from the population. The algorithm can find just the first solution or all. So we can have 10, 20, or 30 different numbers and we will sum some of them to get a desirable number. As an example we have a population of the below numbers: -2,-1,1,2,3,5,8,10 and we try to get 6 - this can be made of 8 and -2, 1 + 5 etc. I need at least 2 decimal places to consider as well for accuracy and ideally, the sum of variables will be exact to the asking value.

Thanks for any advice and help on this:)

I build a model Using the simplex method in Excel but I need the solution in Python.

Celltech
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    You seem to say that your lists of candidates won't get (much) bigger than 30. If this is the case, a brute force approach that just added up every combination would be pretty straightforward to write with a recursive function. If your input list could be significantly larger, then you might need to do something more clever. I'm confident that there are much better ways to do this than the naive brute-force method, but for small-ish lists, such a solution could be fine. Is this a homework assignment? – CryptoFool Nov 12 '22 at 08:51
  • I coded up the brute-force approach. I think I did so very efficiently, but the problem of the combinations you have to check grows so quickly that even lists of 30 or 40 items takes quite a while. So I deleted my answer. If you want to see it, respond to this note, and I'll undelete it, at least long enough for you to check it out. – CryptoFool Nov 12 '22 at 10:18
  • Here's an proper solution to this problem written in Python: https://www.geeksforgeeks.org/python-program-for-subset-sum-problem-dp-25/ – CryptoFool Nov 12 '22 at 10:29

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This is the subset sum problem, which is an NP Complete problem.

There is a known pseudo-polynomial solution for it, if the numbers are integers. In your case, you need to consider numbers only to 2nd decimal point, so you could convert the problem into integers by multiplying by 1001, and then run the pseudo-polynomial algorithm.

It will works quite nicely and efficiently - if the range of numbers you have is quite small (Complexity is O(n*W), where W is the sum of numbers in absolute value).

Appendix:

Pseudo polynomial time solution is Dynamic Programming adaptation of the following recursive formula:

k is the desired number
n is the total number of elements in list.

// stop clause: Found a sum
D(k, i) = true   | for all 0 <= i < n
// Stop clause: failing attempt, cannot find sum in this branch.
D(x, n) = false  | x != k
// Recursive step, either take the current element or skip it.
D(x, i) = D(x + arr[i], i+1) OR D(x, i+1)

Start from D(0,0)

If this is not the case, and the range of numbers is quite high, you might have to go with brute force solution, of checking all possible subsets. This solution is of course exponential, and processing it is in O(2^n) .

(1) Consider rounding if needed, but that's a simple preprocessing that doesn't affect the answer.

amit
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  • Thanks so much for this quite informative and interesting answer! I'm reading over the wiki page you linked above. It really is a very hard problem, isn't it? As I had deduced on my own, bounding the problem to lists of positive integers leads to some much simpler optimizations. So then what if you added the reciprocal of the smallest number to all numbers? Sorry...I'm rambling...thanks again! – CryptoFool Nov 12 '22 at 10:25