preprint
Token swapping on trees
submitted
Ahmad
Biniaz
author
Kshitij
Jain
author
Anna
Lubiw
author
Zuzana
Masárová
author 45CFE238-F248-11E8-B48F-1D18A9856A870000-0002-6660-1322
Tillmann
Miltzow
author
Debajyoti
Mondal
author
Anurag Murty
Naredla
author
Josef
Tkadlec
author 3F24CCC8-F248-11E8-B48F-1D18A9856A87
Alexi
Turcotte
author
HeEd
department
UlWa
department
KrCh
department
The input to the token swapping problem is a graph with vertices v1, v2, . . . , vn, and n tokens with labels 1,2, . . . , n, one on each vertex. The goal is to get token i to vertex vi for all i= 1, . . . , n using a minimum number of 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 NP-complete. 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 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, 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.
ArXiv2019
eng
arXiv:1903.06981
1903.06981
41
https://research-explorer.app.ist.ac.at/record/7944
Biniaz, Ahmad, Kshitij Jain, Anna Lubiw, Zuzana Masárová, Tillmann Miltzow, Debajyoti Mondal, Anurag Murty Naredla, Josef Tkadlec, and Alexi Turcotte. “Token Swapping on Trees.” <i>ArXiv:1903.06981</i>. ArXiv, n.d.
Biniaz A, Jain K, Lubiw A, et al. Token swapping on trees. <i>arXiv:190306981</i>.
Biniaz A, Jain K, Lubiw A, Masárová Z, Miltzow T, Mondal D, Naredla AM, Tkadlec J, Turcotte A. Token swapping on trees. arXiv:1903.06981.
A. Biniaz, K. Jain, A. Lubiw, Z. Masárová, T. Miltzow, D. Mondal, A.M. Naredla, J. Tkadlec, A. Turcotte, ArXiv:1903.06981 (n.d.).
Biniaz, A., Jain, K., Lubiw, A., Masárová, Z., Miltzow, T., Mondal, D., … Turcotte, A. (n.d.). Token swapping on trees. <i>ArXiv:1903.06981</i>. ArXiv.
A. Biniaz <i>et al.</i>, “Token swapping on trees,” <i>arXiv:1903.06981</i>. ArXiv.
Biniaz, Ahmad, et al. “Token Swapping on Trees.” <i>ArXiv:1903.06981</i>, ArXiv.
79502020-06-08T12:25:25Z2020-07-14T23:17:23Z