Token Swapping on Trees
Discrete mathematics & theoretical computer science, Tome 24 (2022) no. 2
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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.
@article{DMTCS_2022_24_2_a10,
author = {Biniaz, Ahmad and Jain, Kshitij and Lubiw, Anna and Mas\'arov\'a, Zuzana and Miltzow, Tillmann and Mondal, Debajyoti and Naredla, Anurag Murty and Tkadlec, Josef and Turcotte, Alexi},
title = {Token {Swapping} on {Trees}},
journal = {Discrete mathematics & theoretical computer science},
year = {2022},
volume = {24},
number = {2},
doi = {10.46298/dmtcs.8383},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.8383/}
}
TY - JOUR AU - Biniaz, Ahmad AU - Jain, Kshitij AU - Lubiw, Anna AU - Masárová, Zuzana AU - Miltzow, Tillmann AU - Mondal, Debajyoti AU - Naredla, Anurag Murty AU - Tkadlec, Josef AU - Turcotte, Alexi TI - Token Swapping on Trees JO - Discrete mathematics & theoretical computer science PY - 2022 VL - 24 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.8383/ DO - 10.46298/dmtcs.8383 LA - en ID - DMTCS_2022_24_2_a10 ER -
%0 Journal Article %A Biniaz, Ahmad %A Jain, Kshitij %A Lubiw, Anna %A Masárová, Zuzana %A Miltzow, Tillmann %A Mondal, Debajyoti %A Naredla, Anurag Murty %A Tkadlec, Josef %A Turcotte, Alexi %T Token Swapping on Trees %J Discrete mathematics & theoretical computer science %D 2022 %V 24 %N 2 %U http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.8383/ %R 10.46298/dmtcs.8383 %G en %F DMTCS_2022_24_2_a10
Biniaz, Ahmad; Jain, Kshitij; Lubiw, Anna; Masárová, Zuzana; Miltzow, Tillmann; Mondal, Debajyoti; Naredla, Anurag Murty; Tkadlec, Josef; Turcotte, Alexi. Token Swapping on Trees. Discrete mathematics & theoretical computer science, Tome 24 (2022) no. 2. doi: 10.46298/dmtcs.8383
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