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Shortest Two Disjoint Paths in Polynomial Time

  • Andreas Björklund
  • Thore Husfeldt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)

Abstract

Given an undirected graph and two pairs of vertices (s i ,t i ) for i ∈ {1,2} we show that there is a polynomial time Monte Carlo algorithm that finds disjoint paths of smallest total length joining s i and t i for i ∈ {1,2} respectively, or concludes that there most likely are no such paths at all. Our algorithm applies to both the vertex- and edge-disjoint versions of the problem.

Our algorithm is algebraic and uses permanents over the quotient ring Z 4[X]/(X m ) in combination with Mulmuley, Vazirani and Vazirani’s isolation lemma to detect a solution. We develop a fast algorithm for permanents over said ring by modifying Valiant’s 1979 algorithm for the permanent over \(\mathbf{Z}_{2^l}\).

Keywords

Polynomial Time Quotient Ring Terminal Vertex Ring Element Laplace Expansion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Andreas Björklund
    • 1
  • Thore Husfeldt
    • 1
    • 2
  1. 1.Lund UniversitySweden
  2. 2.IT Univeristy of CopenhagenDenmark

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