Largest and Smallest Area Triangles on Imprecise Points

TL;DR

This paper investigates the computational complexity of placing points on parallel line segments to optimize the area of the largest or smallest triangles formed, providing polynomial algorithms for some cases and proving NP-hardness for others.

Contribution

It analyzes four triangle area optimization problems on imprecise points, offering polynomial solutions for three and establishing NP-hardness for the remaining case.

Findings

Maximizing the largest triangle in O(n^2) time

Minimizing the largest triangle in O(n^2 log n) time

Maximizing the smallest triangle is NP-hard

Abstract

Assume we are given a set of parallel line segments in the plane, and we wish to place a point on each line segment such that the resulting point set maximizes or minimizes the area of the largest or smallest triangle in the set. We analyze the complexity of the four resulting computational problems, and we show that three of them admit polynomial-time algorithms, while the fourth is NP-hard. Specifically, we show that maximizing the largest triangle can be done in $O(n^2)$ time (or in $O(n \log n)$ time for unit segments); minimizing the largest triangle can be done in $O(n^2 \log n)$ time; maximizing the smallest triangle is NP-hard; but minimizing the smallest triangle can be done in $O(n^2)$ time. We also discuss to what extent our results can be generalized to polygons with $k>3$ sides.