Linear-Time $(1+\varepsilon)$-Approximation Algorithms for Two-Line-Center Problems

TL;DR

This paper introduces efficient $(1+ ext{epsilon})$-approximation algorithms for the two-line-center problem and its variants, achieving linear or near-linear runtime improvements over previous methods for geometric line-fitting tasks.

Contribution

The paper provides the first linear-time approximation algorithms for several variants of the two-line-center problem, significantly improving computational efficiency.

Findings

Achieved $O((n/ ext{epsilon}) ext{log}(1/ ext{epsilon}))$-time algorithm for the main problem.

Developed linear-time $(1+ ext{epsilon})$-approximation algorithms for three problem variants.

Provided an $O(n ext{log} n)$ exact algorithm for a specific fixed-orientation variant.

Abstract

Given a set $S$ of $n$ points in the plane, we study the two-line-center problem: finding two lines that minimize the maximum distance from each point in $S$ to its closest line. We present a $(1+\varepsilon)$-approximation algorithm for the two-line-center problem that runs in $O((n/\varepsilon) \log (1/\varepsilon))$ time, which improves the previously best $O(n\log n + ({n}/{\varepsilon^2}) \log ({1}/{\varepsilon}) + (1/\varepsilon^3)\log ({1}/{\varepsilon}))$-time algorithm. We also consider three variants of this problem, in which the orientations of the two lines are restricted: (1) the orientation of one of the two lines is fixed, (2) the orientations of both lines are fixed, and (3) the two lines are required to be parallel. For each of these three variants, we give the first $(1+\varepsilon)$-approximation algorithm that runs in linear time. In particular, for the variant where the orientation of one of the two lines is fixed, we also give an improved exact algorithm that runs in $O(n \log n)$ time and show that it is optimal.