Partition of Array Problem-NP Complete

the partition problem is an NP-complete problem. The problem is to decide whether a given multiset of integers can be partitioned into two "halves" that have the same sum. More precisely, given a multiset S of integers, is there a way to partition S into two subsets S1 and S2 such that the sum of the numbers in S1 equals the sum of the numbers in S2? The subsets S1 and S2 must form a partition in the sense that they are disjoint and they cover S. The optimization version asks for the "best" partition, and can be stated as: Find a partition into two subsets S1,S2 such that max({sum}(S_1),{sum}(S_2)) is minimized (sometimes with the additional constraint that the sizes of the two sets in the partition must be equal, or differ by at most 1).

In Progress