A generic approach to proving NP-hardness of partition type problems

This note presents a generic approach to proving NP-hardness of unconstrained partition type problems, namely partitioning a given set of entities into several subsets such that a certain objective function of the partition is optimized. The idea is to represent the objective function of the problem as a function of aggregate variables, whose optimum is achieved only at the points where problem Partition (if proving ordinary NP-hardness), or problem 3-Partition or Product Partition (if proving strong NP-hardness) has a solution. The approach is demonstrated on a number of discrete optimization and scheduling problems.