Abstract
In 2003, it was claimed that the following problem was solvable in polynomial time: do there exist k edge-disjoint paths of length exactly 3 between vertices s and t in a given graph? The proof was flawed, and in this note we show that this problem is NP-hard. We use a reduction from Partial Orientation, a problem recently shown by Pálvölgyi to be NP-hard.
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References
Hannah Alpert & Jennifer Iglesias (2012). Length 3 Edge-Disjoint Paths and Partial Orientation. http://arxiv.org/abs/1201.6578v1.
Andreas Bley (2003) On the complexity of vertex-disjoint length-restricted path problems. Comput. Complexity 12(3-4): 131–149 ISSN 1016-3328. doi:10.1007/s00037-003-0179-6
Dömötör Pálvölgyi (2009) Deciding soccer scores and partial orientations of graphs. Acta Univ. Sapientiae Math 1(1): 35–42. ISSN 1844-6094.
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Alpert, H., Iglesias, J. Length 3 Edge-Disjoint Paths Is NP-Hard. comput. complex. 21, 511–513 (2012). https://doi.org/10.1007/s00037-012-0038-4
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DOI: https://doi.org/10.1007/s00037-012-0038-4
Keywords
- Edge-disjoint paths
- NP-hardness
- NP-completeness
- network flow
- directed graph
- oriented graph
- degree sequence