A Quadratic Time Locally Optimal Algorithm for NP-hard Equal Cardinality Partition Optimization
This provides a practical, efficient method for finding locally optimal partitions in various scenarios, though it is incremental as it focuses on a weaker version of the original NP-hard problem.
The paper tackles the NP-hard equal cardinality set partition optimization problem by aiming for a locally optimal solution under pairwise swaps, achieving an algorithm that runs in O(N^2) time and O(N) space without requiring positive or integer inputs.
We study the optimization version of the equal cardinality set partition problem (where the absolute difference between the equal sized partitions' sums are minimized). While this problem is NP-hard and requires exponential complexity to solve in general, we have formulated a weaker version of this NP-hard problem, where the goal is to find a locally optimal solution. The local optimality considered in our work is under any swap between the opposing partitions' element pairs. To this end, we designed an algorithm which can produce such a locally optimal solution in $O(N^2)$ time and $O(N)$ space. Our approach does not require positive or integer inputs and works equally well under arbitrary input precisions. Thus, it is widely applicable in different problem scenarios.