I would like to have a proof of the NP-completeness for 15-Partition Problem.
It is analogous to the well-known 3-Partition Problem. The problem is to decide whether a given multiset of integers can be partitioned into subsets of 15 members that all have the same sum. More precisely, given a multiset S of n = 15 m positive integers, can S be partitioned into m subsets of 15 members S1, S2, …, Sm such that the sum of the numbers in each subset is equal?
It is not an excercise, it is a real problem dealing with load balancing. I did not expect someone would suspect it as a homework/excercise. I mentioned only related facts, obviously, I need to say more. I have a hardware switch with 15 ingoing ports and one outgoing port. Such switches, I have actually 6. That means, I have 15 x 6 = 90 ingoing ports. I have to connect cabels to these 90 ports and I can not affect how much data flows throught these 90 cabels. My task is to assign these 90 cabels to these 90 ports such that the output flow from each switch would be equal. I have already implemented 4 algorithms solving this problem based on dynamic programming, integer linear programming, differential evolution and reinforcement learning.
I would like to have an exact proof that the problem is not so easy to solve, i.e., I would like to prove it belongs to NPC.
Moreover, I have realized today, in case of total flow is not divisible by 6, all switches will not have the equal sum of flow on outgoing port. For example, total flow is 456 254 bytes, then Switch1 to Switch2 should have flow 76 043 bytes each and Switch3 to Switch6 should have flow 76 042 bytes. So maybe, I can not use 15-Partition Problem, not sure about it.