Weight distribution in bucket such that minimum number of buckets are used

Question

I have n weights that weight{A1,A2,...,An} and i have to divide then into buckets such that minimum number of buckets are required and each bucket has maximum capacity of Cmax and Ai<Cmax and size of each bucket should be as close to Cmax as possible Example:- I have weight {1,2,3,4,5} and Cmax= 5 So, result should be ({1,4},{2,3},{5})

{2,3,4,5} and Cmax=10
 So, result should be {{5,3,2},{4}} . 

other possible solution is {5,4},{3,2 } but this is not acceptable as. each initial set should be as much close to Cmax as possible

Answer 1

Sadly, this is NP-hard problem, so exact solution is guaranteed only with brute-force approach - checking all possible variants. It is possible for rather small problem size ( n, buckets < 10..15 ) - there are N^Buckets variants.

Otherwise (usually in case of n < 100 ) you can try some discrete optimization methods like Branch and bound algorithm (stop searching when current solution becomes worse than the best already found one)