Approximating Median Points in a Convex Polygon

TL;DR

This paper presents two efficient approximation algorithms for the continuous k-medians problem in convex polygons, achieving near-optimal solutions with practical performance within 5% to 22% of the optimal.

Contribution

The authors introduce two simple, fast algorithms for continuous k-medians in convex polygons with provable approximation guarantees and practical effectiveness.

Findings

Algorithms run in O(n + k + k log n) time.

Solutions are within a factor of 2.002 of optimal.

Practical tests show solutions within 5% to 22% of optimal.

Abstract

We develop two simple and efficient approximation algorithms for the continuous $k$-medians problems, where we seek to find the optimal location of $k$ facilities among a continuum of client points in a convex polygon $C$ with $n$ vertices in a way that the total (average) Euclidean distance between clients and their nearest facility is minimized. Both algorithms run in $\mathcal{O}(n + k + k \log n)$ time. Our algorithms produce solutions within a factor of 2.002 of optimality. In addition, our simulation results applied to the convex hulls of the State of Massachusetts and the Town of Brookline, MA show that our algorithms generally perform within a range of 5\% to 22\% of optimality in practice.