Question

Suppose that there are n objects with different weights. We need to divide all objects in m sets. Let S_i be sum of i^th set, 1<= i <=m. S_max be maximum of all S_i‘s. What can be the algorithm to minimize S_max over all possible divisions of n objects in m sets.

I have seen some algo which used Hungarian algorithm and bipartite graphs but could not understand that.

Answer 1

The problem you’re asking about is NP-Complete, as it can be reduced to the Partition Problem (and the 3-parition problem). To see why, suppose m = 2 and that the sum of weights of all objects is S. If the first set’s sum is S₁, then obviously the the other set’s sum S₂ = S – S₁. Let’s prove that if we can make S₁ and S₂ equal (the solution to partition problem), then this is the optimal solution for your problem.

If S₁ = S₂ wasn’t optimal, that means that there exists some way of partitioning such that the first set has a sum S₁‘ that would yield a better solution (a smaller maximum number). Let’s consider two cases:

• S₁‘ < S₁. In that case, –S₁‘ > –S₁ and thus S – S₁‘ > S – S₁. Therfore the new maximum is larger (worse) than in the case of equal sums.
• S₁‘ > S₁. In this case, it immediately follows that the maximum in the case of the new partition is larger (worse) than in the case of equal sums.

Similarly, you can show how you can reduce you problem to the 3-paritition problem. However, there are some heuristics that can help in solving the partition problem, you can read about them in the Wikipedia pages I linked to earlier; I doubt there’s anything better than brute force for your more-generic version (unless there’s some kind of restriction you didn’t mention).

Hope that helped you in any manner!