Some Geometric Applications of Anti-Chains

TL;DR

This paper introduces an efficient algorithmic framework for finding maximum anti-chains in geometric posets, enabling polynomial-time solutions for several problems involving points, disks, and rectangles.

Contribution

The paper presents a novel polynomial-time approach to compute maximum anti-chains in geometric posets, solving problems previously considered computationally hard.

Findings

Efficient algorithms for maximum Pareto-optimal points in $\\mathbb{R}^d$

Polynomial-time solution for largest non-containing disk subset

Fast computation of largest non-crossing rectangle subset

Abstract

We present an algorithmic framework for computing anti-chains 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 anti-chain in a poset can be done in polynomial time via maximum-matching 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 Pareto-optimal 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 axis-aligned rectangles, computing the largest subset of non-crossing rectangles.