## Abstract

Given an undirected graph and two disjoint vertex pairs s1 , t1 and s2 , t2 , the Shortest two disjoint paths problem (S2DP) asks for the minimum total length of two vertex disjoint paths connecting s1 with t1, and s2 with t2, respectively.

We show that for cubic planar graphs there are NC algorithms, uniform circuits of polynomial size and polylogarithmic depth, that compute the S2DP and moreover also output the number of such minimum length path pairs.

Previously, to the best of our knowledge, no deterministic polynomial time algorithm was known for S2DP in cubic planar graphs with arbitrary placement of the terminals. In contrast, the randomized polynomial time algorithm by Björklund and Husfeldt, ICALP 2014, for general graphs is much slower, is serial in nature, and cannot count the solutions.

Our results are built on an approach by Hirai and Namba, Algorithmica 2017, for a general- isation of S2DP, and fast algorithms for counting perfect matchings in planar graphs.

We show that for cubic planar graphs there are NC algorithms, uniform circuits of polynomial size and polylogarithmic depth, that compute the S2DP and moreover also output the number of such minimum length path pairs.

Previously, to the best of our knowledge, no deterministic polynomial time algorithm was known for S2DP in cubic planar graphs with arbitrary placement of the terminals. In contrast, the randomized polynomial time algorithm by Björklund and Husfeldt, ICALP 2014, for general graphs is much slower, is serial in nature, and cannot count the solutions.

Our results are built on an approach by Hirai and Namba, Algorithmica 2017, for a general- isation of S2DP, and fast algorithms for counting perfect matchings in planar graphs.

Original language | English |
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Title of host publication | 29th International Symposium on Algorithms and Computation (ISAAC 2018) |

Number of pages | 13 |

Volume | 123 |

Publisher | Schloss Dagstuhl--Leibniz-Zentrum für Informatik |

Publication date | 2018 |

Pages | 19:1-19:13 |

ISBN (Electronic) | ISBN 978-3-95977-094-1 |

DOIs | |

Publication status | Published - 2018 |

Series | Leibniz International Proceedings in Informatics |
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ISSN | 1868-8969 |

## Keywords

- Shortest two disjoint paths
- Cubic planar graphs
- NC algorithms
- Deterministic polynomial time
- Perfect matchings in planar graphs