Approximation and Hardness for Token Swapping

• Main Authors: Miltzow, Tillmann, Narins, Lothar, Okamoto, Yoshio, Rote, Günter, Thomas, Antonis, Uno, Takeaki
• Format: Journal Article
• Language: English
• Published: 02-08-2016
• Subjects: Computer Science - Computational Complexity
• DOI: 10.48550/arxiv.1602.05150

Algorithms-ESA 2016, Proc. 24th Annual European Symposium on
Algorithms, Aarhus, 2016, Leibniz International Proceedings in Informatics
(LIPIcs), pp. 185:1-185:15 Given a graph $G=(V,E)$ with $V=\{1,\ldots,n\}$, we place on every vertex a
token $T_1,\ldots,T_n$. 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.