preprint draft 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-1D18A9856A870000-0002-1097-9684 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. 2019 eng 1903.0698110.48550/arXiv.1903.06981 https://research-explorer.ista.ac.at/record/12833 https://research-explorer.ista.ac.at/record/7944 Biniaz A, Jain K, Lubiw A, et al. Token swapping on trees. <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/arXiv.1903.06981">10.48550/arXiv.1903.06981</a> A. Biniaz, K. Jain, A. Lubiw, Z. Masárová, T. Miltzow, D. Mondal, A.M. Naredla, J. Tkadlec, A. Turcotte, ArXiv (n.d.). Biniaz, Ahmad, et al. “Token Swapping on Trees.” <i>ArXiv</i>, 1903.06981, doi:<a href="https://doi.org/10.48550/arXiv.1903.06981">10.48550/arXiv.1903.06981</a>. Biniaz, A., Jain, K., Lubiw, A., Masárová, Z., Miltzow, T., Mondal, D., … Turcotte, A. (n.d.). Token swapping on trees. <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.1903.06981">https://doi.org/10.48550/arXiv.1903.06981</a> A. Biniaz <i>et al.</i>, “Token swapping on trees,” <i>arXiv</i>. . 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</i>, n.d. <a href="https://doi.org/10.48550/arXiv.1903.06981">https://doi.org/10.48550/arXiv.1903.06981</a>. 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. 79502020-06-08T12:25:25Z2026-04-08T07:23:00Z