eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-08-18
66:1
66:15
10.4230/LIPIcs.ESA.2016.66
article
Approximation and Hardness of Token Swapping
Miltzow, Tillmann
Narins, Lothar
Okamoto, Yoshio
Rote, Günter
Thomas, Antonis
Uno, Takeaki
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol057-esa2016/LIPIcs.ESA.2016.66/LIPIcs.ESA.2016.66.pdf
token swapping
minimum generator sequence
graph theory
NP-hardness
approximation algorithms