Flipping Matchings is Hard

TL;DR

The paper addresses the problem of transforming one plane perfect matching into another on a fixed point set using a sequence of edge flips, formalized as FlippingBetweenMatchings. It introduces a gadget-based NP-hardness reduction from Planar Vertex Cover, establishing that a yes instance with budget $k$ exists exactly when a vertex cover of size at most $c$ exists, via a construction with vertex gadgets, edge gadgets, blockers, separators, and a budget $k=2c+5|E|$. The core insight is that activating a subset of vertex gadgets corresponding to a vertex cover suffices to reconfigure $ ext{M}_1$ into $ ext{M}_2$ in the specified number of flips, and any feasible sequence encodes a vertex cover of size $oxed{c}$. The result holds for integer-coordinate point sets with polynomial-area bounding box, highlighting a fundamental hardness barrier for reconfiguring plane perfect matchings and prompting further work on general-position instances and flip-graph connectivity.

Abstract

Given a point set $\mathcal{P}$ and a plane perfect matching $\mathcal{M}$ on $\mathcal{P}$, a flip is an operation that replaces two edges of $\mathcal{M}$ such that another plane perfect matching on $\mathcal{P}$ is obtained. Given two plane perfect matchings on $\mathcal{P}$, we show that it is NP-hard to minimize the number of flips that are needed to transform one matching into the other.

Figures (10)

• Figure 1: Flipping edges $e_1$ and $e_2$ to $e_3$ and $e_4$; points are drawn as yellow circles.
• Figure 2: (a) A graph $G$ where a vertex cover of size $3$ is depicted in gray. (b) A weak-visibility representation $R$ of $G$. (c) The plane perfect matching $\mathcal{M}_1\xspace$ obtained by replacing each vertex-segment of $R$ by a vertex gadget and each edge-segment of $R$ by an edge gadget. (d) The plane perfect matching $\mathcal{M}_2\xspace$. Each bold line represents $2k+2$ line segments. The vertex $v_3$ is red and the edge $v_1v_3$ is blue in each representation.
• Figure 3: Edge gadget in the start-configuration.
• Figure 4: Edge gadget in the final-configuration.
• Figure 6: Deactivated vertex gadget. The bold edges are a set of $2k+2$ edges.

Theorems & Definitions (19)

• theorem 1
• lemma 1
• proof
• lemma 2
• proof : Proof sketch
• lemma 3
• proof
• lemma 4
• proof
• lemma 5