Optimizing an Algorithm for Timestamp-Aware Partitioning of Data

Question

My Problem

I'm currently dealing with an algorithmic problem that involves two input lists:

1. A list of natural numbers $[A_1, A_2, \dots, A_n]$ with $A_1, \dots, A_n \in \mathbb{N}$.
2. A list of triples $[(B_1, C_1, D_1), (B_2, C_2, D_2), \dots, (B_m, C_m, D_m)]$ with $m\geq n$, natural numbers $B_1, \dots, B_m \in \mathbb{N}$, attributes $C_1, \dots, C_m$ and timestamps $D_1, \dots, D_m$.

Let $\mathcal{X}$ represent the set of triples in the second list. My objective is to find a partition $(\mathcal{P}_1, \dots, \mathcal{P}_n)$ of $\mathcal{X}$ that satisfies the following conditions:

1. The sum of natural numbers in the $i$-th partition must equal $A_i$. In other words, $\sum_{(B,C,D) \in \mathcal{P}_i} B = A_i$ for all $i = 1, \dots, n$.
2. Only triples sharing the same attribute $C$ should be placed in the same set within the partition. This means that for all $i\in 1,\dots, n$, there exists some $c_i$ such that $C = c_i$ for all $(B,C,D) \in \mathcal{P}_i$.
3. Among all partitions satisfying condition 1 and 2, we aim to minimize the timestamp difference within the sets of the partition. To formalize this, let's denote the time difference between two timestamps $D$ and $D^\prime$ as $d(D, D^\prime)$. Our goal is to minimize the following expression

$$ \sum_{i=1,\dots, n} \max_{(B,C,D), (B^\prime, C^\prime, D^\prime) \in \mathcal{P}_i} d(D, D^\prime). $$

We can assume that a solution exists.

My Approach and Objective

So far, I've developed an algorithm that involves exploring various combinations of partitions while minimizing the number of operations required to verify whether a partition is promising. I've dedicated a significant amount of time in search of a more sophisticated algorithm that doesn't rely on trying every combination sequentially.

I'm looking for assistance and guidance in finding a faster algorithm. Any insights or suggestions would be greatly appreciated. :)