Flipping Matchings is Hard

2025-03-04

AI Summary

This paper investigates the minimum flip distance between two plane perfect matchings on a planar point set, where each flip replaces two edges in the matching to yield another plane perfect matching. The authors establish the NP-hardness of this problem via a constructive reduction from Planar 3SAT—a known NP-complete problem—leveraging planar graph embedding properties and structural constraints of plane matchings to ensure correctness. This is the first rigorous proof of NP-hardness for the plane perfect matching flip distance problem. The result reveals the intrinsic computational intractability of optimizing local transformation sequences in combinatorial geometry, ruling out the existence of polynomial-time exact algorithms. It provides foundational theoretical insights for problems involving matching deformation, geometric reconfiguration, and structural analysis of flip graphs—combinatorial graphs whose vertices represent plane perfect matchings and whose edges correspond to single flips.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• NP-hardness of minimizing flips
• Transforming plane perfect matchings
• Flip operations on point sets

Innovation

• Flip operation transforms plane perfect matchings
• NP-hard to minimize flip count
• Focus on point set and matching transformations