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 \(\mathbf{Z}_{2^l}\).
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Björklund, A., Husfeldt, T. (2014). Shortest Two Disjoint Paths in Polynomial Time. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds) Automata, Languages, and Programming. ICALP 2014. Lecture Notes in Computer Science, vol 8572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43948-7_18
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DOI: https://doi.org/10.1007/978-3-662-43948-7_18
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