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Let me give you an example of a variation on the normal subset-sum problem that I am trying to solve:

Given is a set S = {1,2,3,4,5,6,7,8,9} with the maximal capacity c0 = 40. Furthermore we have 3 additional constraints on 3 different subsets of S:

  1. S1 = {2,3,4} with constraint c1 = 5
  2. S2 = {3,4,5,6} with constraint c2 = 12
  3. S3 = {7,8,9} with constraint c3 = 25

The objective is to find a subset of S such that the total sum (of the included items) is maximized without exceeding ANY given constraint (c0 - c4)

Important:

  • intersections are possible! (see S1 & S2)
  • 3 is just an example for the count of constraints - it could be considerably more!
  • although the items of S are integer values in this example, it could also be positive real numbers

Question: Does this particular subset-sum problem have a specific name and/or are there any papers/ literature reviews on this ?

Arusa
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This sounds like the knapsack problem: https://en.wikipedia.org/wiki/Knapsack_problem

See the section in that article on the subset sum problem:

"The subset sum problem is a special case of the decision and 0-1 problems where each kind of item, the weight equals the value[...] In the field of cryptography, the term knapsack problem is often used to refer specifically to the subset sum problem and is commonly known as one of Karp's 21 NP-complete problems.[30]

The generalization of subset sum problem is called multiple subset-sum problem, in which multiple bins exist with the same capacity. It has been shown that the generalization does not have an FPTAS.[31]"

Andrew
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