# Shortest Reconfiguration of Matchings

## TL;DR

This paper investigates the shortest reconfiguration sequences between matchings in graphs, providing polynomial algorithms, complexity results, and parameterized approaches, with applications to bipartite graphs and connections to the Directed Steiner Tree problem.

## Contribution

It introduces polynomial-time algorithms for certain cases, proves NP-hardness and inapproximability results, and develops fixed-parameter tractable and approximation algorithms for matching reconfiguration.

## Key findings

- Polynomial-time algorithm when at least one configuration is not maximal.
- NP-hardness and no sublogarithmic approximation unless P=NP.
- Fixed-parameter tractability and approximation algorithms for bipartite graphs.

## Abstract

Imagine that unlabelled tokens are placed on the edges of a graph, such that no two tokens are placed on incident edges. A token can jump to another edge if the edges having tokens remain independent. We study the problem of determining the distance between two token configurations (resp., the corresponding matchings), which is given by the length of a shortest transformation. We give a polynomial-time algorithm for the case that at least one of the two configurations is not inclusion-wise maximal and show that otherwise, the problem admits no polynomial-time sublogarithmic-factor approximation unless P = NP. Furthermore, we show that the distance of two configurations in bipartite graphs is fixed-parameter tractable parameterized by the size $d$ of the symmetric difference of the source and target configurations, and obtain a $d^\varepsilon$-factor approximation algorithm for every $\varepsilon > 0$ if additionally the configurations correspond to maximum matchings. Our two main technical tools are the Edmonds-Gallai decomposition and a close relation to the Directed Steiner Tree problem. Using the former, we also characterize those graphs whose corresponding configuration graphs are connected. Finally, we show that deciding if the distance between two configurations is equal to a given number $\ell$ is complete for the class $D^P$, and deciding if the diameter of the graph of configurations is equal to $\ell$ is $D^P$-hard.