eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-09-01
3:1
3:15
10.4230/LIPIcs.ESA.2022.3
article
Hardness of Token Swapping on Trees
Aichholzer, Oswin
1
Demaine, Erik D.
2
https://orcid.org/0000-0003-3803-5703
Korman, Matias
3
Lubiw, Anna
4
https://orcid.org/0000-0002-2338-361X
Lynch, Jayson
4
Masárová, Zuzana
5
Rudoy, Mikhail
6
Vassilevska Williams, Virginia
2
Wein, Nicole
7
Technische Universität Graz, Austria
CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Siemens Electronic Design Automation, Wilsonville, OR, USA
Cheriton School of Computer Science, University of Waterloo, Canada
IST Austria, Klosterneuburg, Austria
LeapYear Technologies, San Francisco, CA, USA
DIMACS, Rutgers University, Piscataway, NJ, USA
Given a graph where every vertex has exactly one labeled token, how can we most quickly execute a given permutation on the tokens? In (sequential) token swapping, the goal is to use the shortest possible sequence of swaps, each of which exchanges the tokens at the two endpoints of an edge of the graph. In parallel token swapping, the goal is to use the fewest rounds, each of which consists of one or more swaps on the edges of a matching. We prove that both of these problems remain NP-hard when the graph is restricted to be a tree.
These token swapping problems have been studied by disparate groups of researchers in discrete mathematics, theoretical computer science, robot motion planning, game theory, and engineering. Previous work establishes NP-completeness on general graphs (for both problems), constant-factor approximation algorithms, and some poly-time exact algorithms for simple graph classes such as cliques, stars, paths, and cycles. Sequential and parallel token swapping on trees were first studied over thirty years ago (as "sorting with a transposition tree") and over twenty-five years ago (as "routing permutations via matchings"), yet their complexities were previously unknown.
We also show limitations on approximation of sequential token swapping on trees: we identify a broad class of algorithms that encompass all three known polynomial-time algorithms that achieve the best known approximation factor (which is 2) and show that no such algorithm can achieve an approximation factor less than 2.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol244-esa2022/LIPIcs.ESA.2022.3/LIPIcs.ESA.2022.3.pdf
Sorting
Token swapping
Trees
NP-hard
Approximation