In MDK, we have a vector $W = \{W_1, W_2, ..., W_d\}$ where each element corresponds to the maximum weight for the respective dimension in the knapsack.
I want to add a conditional constraint: $V = {V_1, V_2, ..., V_d}$, where each $i$-th dimension in the knapsack must have a value sum greater than threshold $V_i$. I am not so much concerned with the total value sum.
I would like to show this problem is NP-hard. My intuition is that the additional constraint makes this problem harder than MKD and therefore is NP-hard. But clearly this doesn't constitute a formal proof.