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

2026-01-07

AI Summary

This work addresses the two-line center problem: given $n$ points in the plane, find two lines that minimize the maximum distance from any point to its nearest line. The paper presents the first linear-time $(1+\varepsilon)$-approximation algorithms for the general case and three constrained variants—where one line’s direction is fixed, both directions are fixed, or the two lines are parallel. The algorithm for the general case runs in $O((n/\varepsilon)\log(1/\varepsilon))$ time, significantly improving upon prior results. For the single-fixed-direction variant, an optimal exact algorithm with $O(n\log n)$ time complexity is developed, matching the theoretical lower bound. The approach integrates geometric approximation, divide-and-conquer strategies, parametric search, and directional discretization techniques.

Technology Category

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• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

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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.

Problem

• two-line-center
• approximation algorithm
• computational geometry
• line orientation
• minimax distance

Innovation

• two-line-center
• linear-time approximation
• geometric optimization
• direction-constrained lines
• computational geometry