Given an array $A=\{\frac{S_1}{K_1},\frac{S_2}{K_2},...,\frac{S_n}{K_n}\}$ whose any $S_i$ and $K_i$ are positive floating-point, and an positive floating-point $R$, how to select a set of element $C=\{\frac{S_{p_1}}{K_{p_1}},\frac{S_{p_2}}{K_{p_2}},...,\frac{S_{p_m}}{K_{p_m}}\}$ such that the absolute value of the difference between $\frac{\sum_{i=1}^m S_{p_i}}{\sum_{i=1}^m K_{p_i}}$ and $R$ (i.e. $|\frac{\sum_{i=1}^m S_{p_i}}{\sum_{i=1}^m K_{p_i}}-R|$)is minimal. The subset $C$ has at least one element.
This problem has confused me for many weeks. I can't even prove whether it is an NP or NP-hard problem. Could anyone give me some ideas on how to address it? I really appreciate any help you can provide.

  • $\begingroup$ What's the context where you encountered it? Can you credit the original source? $\endgroup$
    – D.W.
    Nov 15, 2022 at 7:36
  • $\begingroup$ This problem is abstracted from a model we are studying. The model is actually a scheduling decision model for heterogeneous computing. We tried to tackle it for many weeks but got very little progress (probably due to our limited algorithm addressing ability). $\endgroup$
    – Virux
    Nov 15, 2022 at 7:58

1 Answer 1


The special case where $K_1=K_2=\cdots=K_n$ is as hard as the subset-sum problem, which is NP-hard -- so your problem is NP-hard as well.

A plausible approach is to use integer linear programming. Let $x_i$ be a zero-or-one integer variable. You are looking for $x_i$ such that

$$-\epsilon \sum_i x_i K_i \le \sum_i x_i S_i - R \sum_i x_i K_i \le \epsilon \sum_i x_i K_i.$$

For fixed $\epsilon$, you can test whether there is a solution to this linear inequality using an ILP solver. Then, you can use binary search on $\epsilon$ to find the smallest $\epsilon$ for which a solution exists.


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