### Abstract

Consider a puzzle consisting of n tokens on an n-vertex graph, where each token has a distinct starting vertex and a distinct target vertex it wants to reach, and the only allowed transformation is to swap the tokens on adjacent vertices. We prove that every such puzzle is solvable in O(n^{2}) token swaps, and thus focus on the problem of minimizing the number of token swaps to reach the target token placement. We give a polynomial-time 2-approximation algorithm for trees, and using this, obtain a polynomial-time 2α-approximation algorithm for graphs whose tree α-spanners can be computed in polynomial time. Finally, we show that the problem can be solved exactly in polynomial time on complete bipartite graphs.

Original language | English |
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Pages (from-to) | 81-94 |

Number of pages | 14 |

Journal | Theoretical Computer Science |

Volume | 586 |

DOIs | |

Publication status | Published - 2015 Jun 27 |

### Keywords

- Approximation
- Complete bipartite graph
- Graph algorithm
- Sorting network
- Tree

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Theoretical Computer Science*,

*586*, 81-94. https://doi.org/10.1016/j.tcs.2015.01.052