# NP-hardness of minimizing sum of complicated objective function

In our research, we faced the following problem optimization problem:

Input: a list of $k$ pairs of positive integers $(n_1,d_1), \ldots, (n_k,d_k)$; an integer $m$.
Output: $P$, a partition of the pairs into $m$ nonempty classes,
such that the following objective function is minimized:

$$\sum_{C \in P}\left( \left(\sum_{i \in C} d_i\hspace{-0.08 in}\right) \cdot \left(\sqrt{\frac{ 2\log(4\sum_{i\in C} d_i)}{\sum_{i \in C}n_i}} + \sqrt{\frac{ \log(2\sum_{i\in C} d_i)}{\sum_{i \in C}n_i}}\hspace{.04 in}\right)\hspace{-0.04 in} \right)$$ We think that the problem is NP-hard, but we have not been able to prove this.
We tried to prove that it is NP-hard by reducing it to 3-Partition, but we couldn't.

Is this problem NP-hard?

• Note: Accepted edit since I couldn't view the effects otherwise. Rollbacked because the change did not improve the formula and the second part did not make sense. See edit history for details. Commented Jul 28, 2015 at 23:37
• I think the OP means "complex" as "not simple", not as "not real" Commented Jul 29, 2015 at 3:34
• I don't understand what the inner summation $\sum d_i$ is over Commented Jul 29, 2015 at 3:36
• The formulation is wrong. Tried to correct, but was not able to do it. The correct one is $\sum_{C_j}\left(\left(\sum_{I_i \in C_j}d_i\right)\left(\sqrt{\frac{2\log(4\sum_{I_i \in C_j}d_i)}{\sum_{I_i \in C_j}n_i}} + \sqrt{\frac{\log(2\sum_{I_i \in C_j}d_i)}{\sum_{I_i \in C_j}n_i}}\right)\right)$ Hope this formula makes sense. Commented Jul 29, 2015 at 17:03