# The balanced 2-median and 2-maxian problems on a tree

## TL;DR

This paper introduces new algorithms for balanced 2-median and 2-maxian problems on trees, focusing on balancing demand while optimizing location objectives, with efficient solutions for both problems.

## Contribution

It presents an O(n^2) edge deletion method for the balanced 2-median problem and a linear-time solution for the balanced 2-maxian problem on trees.

## Key findings

- Balanced 2-median problem solvable in O(n^2) time.
- Optimal 2-maxian solution is two diameter endpoints, found in linear time.
- Proposes bi-objective models combining location and demand balancing.

## Abstract

This paper deals with the facility location problems with balancing on allocation clients to servers. Two bi-objective models are considered, in which one objective is the traditional p-median or p-maxian objective and the second is to minimize the maximum demand volume allocated to any facility. An edge deletion method with time complexity O(n^2) is presented for the balanced $2$-median problem on a tree. For the balanced 2-maxian problem, it is shown the optimal solution is two end vertices of the diameter of the tree, which can be obtained in a linear time.