Token Swapping on Trees
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 \emph{swaps}, where a swap exchanges the tokens on the endpoints of an edge. Token swapping on a tree, also known as "sorting with a transposition tree", is not known to be in P nor NPcomplete. We present some partial results: 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 leavesas all known approximation algorithms for the problem dohas approximation factor at least $4/3$. Furthermore, the two bestknown 2approximation algorithms have approximation factor exactly 2. 3. A generalized problemweighted coloured token swappingis NPcomplete on trees, 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2019
 arXiv:
 arXiv:1903.06981
 Bibcode:
 2019arXiv190306981B
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms;
 03D15;
 05C05;
 68R05