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Hardness of Token Swapping on Trees

Authors Oswin Aichholzer, Erik D. Demaine , Matias Korman, Anna Lubiw , Jayson Lynch, Zuzana Masárová, Mikhail Rudoy, Virginia Vassilevska Williams, Nicole Wein

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Author Details

Oswin Aichholzer
  • Technische Universität Graz, Austria
Erik D. Demaine
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Matias Korman
  • Siemens Electronic Design Automation, Wilsonville, OR, USA
Anna Lubiw
  • Cheriton School of Computer Science, University of Waterloo, Canada
Jayson Lynch
  • Cheriton School of Computer Science, University of Waterloo, Canada
Zuzana Masárová
  • IST Austria, Klosterneuburg, Austria
Mikhail Rudoy
  • LeapYear Technologies, San Francisco, CA, USA
Virginia Vassilevska Williams
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Nicole Wein
  • DIMACS, Rutgers University, Piscataway, NJ, USA

Funding

  • Lubiw, Anna: Supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).
  • Lynch, Jayson: Supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).
  • Masárová, Zuzana: Supported by Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.
  • Vassilevska Williams, Virginia: Supported by an NSF CAREER Award, NSF Grants CCF-1528078, CCF-1514339 and CCF-1909429, a BSF Grant BSF:2012338, a Google Research Fellowship and a Sloan Research Fellowship.
  • Wein, Nicole: Supported by a grant to DIMACS from the Simons Foundation (820931). This work was done while the author was at MIT.

Acknowledgements

This research was initiated at the 34th Bellairs Winter Workshop on Computational Geometry, co-organized by Erik Demaine and Godfried Toussaint, held on March 22-29, 2019 in Holetown, Barbados. We thank the other participants of that workshop for providing a stimulating research environment.

Cite AsGet BibTex

Oswin Aichholzer, Erik D. Demaine, Matias Korman, Anna Lubiw, Jayson Lynch, Zuzana Masárová, Mikhail Rudoy, Virginia Vassilevska Williams, and Nicole Wein. Hardness of Token Swapping on Trees. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ESA.2022.3

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Sorting
  • Token swapping
  • Trees
  • NP-hard
  • Approximation

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References

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