Some Geometric Applications of AntiChains
Abstract
We present an algorithmic framework for computing antichains of maximum size in geometric posets. Specifically, posets in which the entities are geometric objects, where comparability of two entities is implicitly defined but can be efficiently tested. Computing the largest antichain in a poset can be done in polynomial time via maximummatching in a bipartite graph, and this leads to several efficient algorithms for the following problems, each running in (roughly) $O(n^{3/2})$ time: (A) Computing the largest Paretooptimal subset of a set of $n$ points in $\mathbb{R}^d$. (B) Given a set of disks in the plane, computing the largest subset of disks such that no disk contains another. This is quite surprising, as the independent version of this problem is computationally hard. (C) Given a set of axisaligned rectangles, computing the largest subset of noncrossing rectangles.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 DOI:
 10.48550/arXiv.1910.07586
 arXiv:
 arXiv:1910.07586
 Bibcode:
 2019arXiv191007586H
 Keywords:

 Computer Science  Computational Geometry
 EPrint:
 To appear in CCCG 2020