A Quadratic Time Locally Optimal Algorithm for NP-hard Equal Cardinality Partition Optimization

TL;DR

This paper introduces a quadratic time algorithm for finding locally optimal solutions to the NP-hard equal cardinality partition problem, applicable to arbitrary inputs and designed for efficiency.

Contribution

The authors present a novel $O(N^2)$ time algorithm that efficiently finds locally optimal solutions for the equal cardinality partition problem, broadening applicability beyond positive integers.

Findings

Algorithm runs in $O(N^2)$ time and $O(N)$ space.

Works with arbitrary input precisions, not limited to positive integers.

Produces locally optimal solutions efficiently for large instances.

Abstract

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.