TY - GEN

T1 - Hardness of Token Swapping on Trees

AU - Aichholzer, Oswin

AU - Demaine, Erik D.

AU - Korman, Matias

AU - Lubiw, Anna

AU - Lynch, Jayson

AU - Masárová, Zuzana

AU - Rudoy, Mikhail

AU - Williams, Virginia Vassilevska

AU - Wein, Nicole

N1 - Publisher Copyright:
© 2022 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

PY - 2022/9/1

Y1 - 2022/9/1

N2 - 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.

AB - 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.

KW - Approximation

KW - NP-hard

KW - Sorting

KW - Token swapping

KW - Trees

UR - http://www.scopus.com/inward/record.url?scp=85137546139&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2022.3

DO - 10.4230/LIPIcs.ESA.2022.3

M3 - Conference paper

AN - SCOPUS:85137546139

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 30th Annual European Symposium on Algorithms, ESA 2022

A2 - Chechik, Shiri

A2 - Navarro, Gonzalo

A2 - Rotenberg, Eva

A2 - Herman, Grzegorz

PB - Schloss Dagstuhl - Leibniz-Zentrum für Informatik

T2 - 30th Annual European Symposium on Algorithms

Y2 - 5 September 2022 through 9 September 2022

ER -