# Token Swapping on Trees

## TL;DR

This paper investigates the token swapping problem on trees, revealing new complexities, disproving previous conjectures, and analyzing approximation algorithms and their limitations.

## Contribution

It provides new insights into token swapping on trees, including counterexamples to conjectures, approximation bounds, and complexity results for weighted coloured variants.

## Key findings

- Optimal swaps may involve happy leaves, countering previous assumptions.
- Approximation algorithms have a lower bound of 4/3, with the best known algorithms achieving a factor of 2.
- Weighted coloured token swapping is NP-complete on trees but polynomial on paths and stars.

## Abstract

The input to the token swapping problem is a graph with vertices $v_1, v_2, \ldots, v_n$, and $n$ tokens with labels $1, 2, \ldots, n$, one on each vertex. The goal is to get token $i$ to vertex $v_i$ for all $i= 1, \ldots, n$ using a minimum number of swaps, where a swap exchanges the tokens on the endpoints of an edge. We present some results about token swapping on a tree, also known as "sorting with a transposition tree": 1. An optimum swap sequence may need to perform a swap on a leaf vertex that has the correct token (a "happy leaf"), disproving a conjecture of Vaughan. 2. Any algorithm that fixes happy leaves -- as all known approximation algorithms for the problem do -- has approximation factor at least $4/3$. Furthermore, the two best-known 2-approximation algorithms have approximation factor exactly 2. 3. A generalized problem -- weighted coloured token swapping -- is NP-complete on trees, even when they are restricted to be subdivided stars, but solvable in polynomial time on paths and stars. In this version, tokens and vertices have colours, and colours have weights. The goal is to get every token to a vertex of the same colour, and the cost of a swap is the sum of the weights of the two tokens involved.