Efficient Locally Optimal Number Set Partitioning for Scheduling, Allocation and Fair Selection

TL;DR

This paper introduces a near-linear time algorithm for finding locally optimal solutions to a weaker, more applicable version of the NP-hard set partition problem, with broad potential applications.

Contribution

The paper proposes a novel, efficient algorithm for locally optimal set partitioning that works on general input sets and is computationally faster than traditional methods.

Findings

Algorithms find locally optimal solutions in near-linear time.

Applicable to general sets without positive or integer constraints.

Offers a practical approach for large-scale decision problems.

Abstract

We study the optimization version of the set partition problem (where the difference between the partition sums are minimized), which has numerous applications in decision theory literature. While the set partitioning problem is NP-hard and requires exponential complexity to solve (i.e., intractable); we formulate a weaker version of this NP-hard problem, where the goal is to find a locally optimal solution. We show that our proposed algorithms can find a locally optimal solution in near linear time. Our algorithms require neither positive nor integer elements in the input set, hence, they are more widely applicable.