Complexity Results on Untangling Red-Blue Matchings

TL;DR

This paper investigates the complexity of untangling red-blue matchings in the plane through flip operations, proving NP-hardness for approximation and providing bounds and constructions for specific configurations.

Contribution

It establishes NP-hardness for approximating shortest flip sequences and provides new bounds and constructions for red-blue matchings in colinear and convex cases.

Findings

NP-hard to alpha-approximate shortest flip sequence

Existence of flip sequences of length at most n(n-1)/2 in colinear case

New bounds on flip sequence lengths and constructions in convex and colinear cases

Abstract

Given a matching between n red points and n blue points by line segments in the plane, we consider the problem of obtaining a crossing-free matching through flip operations that replace two crossing segments by two non-crossing ones. We first show that (i) it is NP-hard to alpha-approximate the shortest flip sequence, for any constant alpha. Second, we show that when the red points are colinear, (ii) given a matching, a flip sequence of length at most n(n-1)/2 always exists, and (iii) the number of flips in any sequence never exceeds (n(n-1)/2) (n+4)/6. Finally, we present (iv) a lower bounding flip sequence with roughly 1.5 n(n-1)/2 flips, which shows that the n(n-1)/2 flips attained in the convex case are not the maximum, and (v) a convex matching from which any flip sequence has roughly 1.5 n flips. The last four results, based on novel analyses, improve the constants of state-of-the-art bounds.