Some Geometric Applications of Anti-Chains

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.

💡 Research Summary

The paper introduces a unified algorithmic framework for finding a maximum‑size anti‑chain in geometric partially ordered sets (posets) where the elements are geometric objects and the comparability relation can be tested efficiently. The authors observe that computing a largest anti‑chain is equivalent to finding a maximum matching in a bipartite graph derived from the poset, a classic consequence of Dilworth’s theorem. While the standard Hopcroft–Karp algorithm solves the matching in O(|E|·√|V|) time, the paper shows how to replace the explicit construction of the edge set with on‑the‑fly neighbor queries using dynamic range‑search data structures. If each neighbor query and deletion can be performed in T(m) time for a set of size m, the whole matching can be obtained in O(n^{1.5}·T(n)) time.

The framework rests on two modest assumptions: (i) comparability of any two elements can be decided in constant time, and (ii) given an element v, one can quickly locate an element u such that v ≺ u, using a data structure that supports queries and deletions. Under these conditions the authors obtain O(n^{1.5})‑type algorithms for several geometric problems:

1. Maximum Pareto‑optimal subset in ℝ^d – Points are ordered by coordinate‑wise domination (p ≽ q). The query “find a point that dominates q” becomes a d‑sided orthogonal range query on the region