Token swapping on trees
Biniaz, Ahmad
Jain, Kshitij
Lubiw, Anna
Masárová, Zuzana
Miltzow, Tillmann
Mondal, Debajyoti
Naredla, Anurag Murty
Tkadlec, Josef
Turcotte, Alexi
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
info:eu-repo/semantics/preprint
doc-type:preprint
text
http://purl.org/coar/resource_type/c_816b
https://research-explorer.ista.ac.at/record/7950
Biniaz A, Jain K, Lubiw A, et al. Token swapping on trees. <i>arXiv</i>.
eng
info:eu-repo/semantics/altIdentifier/arxiv/1903.06981
info:eu-repo/semantics/openAccess