Given a graph G=(V,E) with V={1,...,n}, we place on every vertex a token T_1,...,T_n. A swap is an exchange of tokens on adjacent vertices. We consider the algorithmic question of finding a shortest sequence of swaps such that token T_i is on vertex i. We are able to achieve essentially matching upper and lower bounds, for exact algorithms and approximation algorithms. For exact algorithms, we rule out any 2^{o(n)} algorithm under the ETH. This is matched with a simple 2^{O(n*log(n))} algorithm based on a breadth-first search in an auxiliary graph. We show one general 4-approximation and show APX-hardness. Thus, there is a small constant delta > 1 such that every polynomial time approximation algorithm has approximation factor at least delta. Our results also hold for a generalized version, where tokens and vertices are colored. In this generalized version each token must go to a vertex with the same color.
@InProceedings{miltzow_et_al:LIPIcs.ESA.2016.66, author = {Miltzow, Tillmann and Narins, Lothar and Okamoto, Yoshio and Rote, G\"{u}nter and Thomas, Antonis and Uno, Takeaki}, title = {{Approximation and Hardness of Token Swapping}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {66:1--66:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.66}, URN = {urn:nbn:de:0030-drops-64084}, doi = {10.4230/LIPIcs.ESA.2016.66}, annote = {Keywords: token swapping, minimum generator sequence, graph theory, NP-hardness, approximation algorithms} }
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