# Multiple knapsack problem with equal profit and different weight

I am doing a research about the load balancing problem in 5G system, but I am not sure if my problem is a NP-complete problem.

The problem is:

• given a set of n items and a set of m knapsack
• capacity of knapsacks are equal
• the weight of item j in knapsack i is w[i][j],that means weight of a item in each knapsack are different
• each profit of items are equal

I am not trying to put all item in least knapsack like bin packing problem. I saw some similar question answered, but no one is identical to this case. In this case, the goal is to put as more as possible item with m knapsacks. Is the problem a NP-complete problem?

This problem can be shown to be NP-complete via reduction from PARTITION. Simply take $$m=2$$, the weights of each item to be the same across both knapsacks, and the capacities of each knapsack to be half the total weight across all items.