Faster Distance-Based Representative Skyline and $k$-Center Along Pareto Front in the Plane

TL;DR

This paper introduces faster algorithms for computing distance-based representative skylines and $k$-center problems along Pareto fronts in the plane, significantly improving efficiency over previous methods.

Contribution

It presents new algorithms with optimal or near-optimal time complexities for computing representative skylines and $k$-center problems in multi-objective optimization.

Findings

Achieves $O(n ext{log}h)$ time for the main problem

Decision problem solved in $O(n ext{log}k)$ time

Optimization problem solved in $O(n ext{log}k + n ext{loglog}n)$ time

Abstract

We consider the problem of computing the \emph{distance-based representative skyline} in the plane, a problem introduced by Tao, Ding, Lin and Pei [Proc. 25th IEEE International Conference on Data Engineering (ICDE), 2009] and independently considered by Dupin, Nielsen and Talbi [Optimization and Learning - Third International Conference, OLA 2020] in the context of multi-objective optimization. Given a set $P$ of $n$ points in the plane and a parameter $k$, the task is to select $k$ points of the skyline defined by $P$ (also known as Pareto front for $P$) to minimize the maximum distance from the points of the skyline to the selected points. We show that the problem can be solved in $O(n\log h)$ time, where $h$ is the number of points in the skyline of $P$. We also show that the decision problem can be solved in $O(n\log k)$ time and the optimization problem can be solved in $O(n \log k + n \operatorname{loglog} n)$ time. This improves previous algorithms and is optimal for a large range of values of $k$.