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?
NP-hardness of minimizing sum of complicated objective function
cc.complexity theory;np hardness;optimization;partition problem