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On Finding Min-Min Disjoint Paths

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Abstract

The Min-Min problem of finding a disjoint-path pair with the length of the shorter path minimized is known to be NP-hard and admits no K-approximation for any K>1 in the general case (Xu et al. in IEEE/ACM Trans. Netw. 14:147–158, 2006).

In this paper, we first show that Bhatia et al.’s NP-hardness proof (Bhatia et al. in J. Comb. Optim. 12:83–96, 2006), a claim of correction to Xu et al.’s proof (Xu et al. in IEEE/ACM Trans. Netw. 14:147–158, 2006), for the edge-disjoint Min-Min problem in the general undirected graphs is incorrect by giving a counter example that is an unsatisfiable 3SAT instance but classified as a satisfiable 3SAT instance in the proof of Bhatia et al. (J. Comb. Optim. 12:83–96, 2006). We then gave a correct proof of NP-hardness of this problem in undirected graphs. Finally we give a polynomial-time algorithm for the vertex disjoint Min-Min problem in planar graphs by showing that the vertex disjoint Min-Min problem is polynomially solvable in st-planar graph G=(V,E) whose corresponding auxiliary graph G(V,E∪{e(st)}) can be embedded into a plane, and a planar graph can be decomposed into several st-planar graphs whose Min-Min paths collectively contain a Min-Min disjoint-path pair between s and t in the original graph G. To the best of our knowledge, these are the first polynomial algorithms for the Min-Min problems in planar graphs.

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Authors and Affiliations

  1. School of Mathematics and Computer Science, Fuzhou University, Fuzhou, China

    Longkun Guo

  2. School of Computer and Information Technology, Beijing Jiaotong University, Beijing, China

    Hong Shen

  3. School of Computer Science, University of Adelaide, Adelaide, Australia

    Hong Shen

Authors
  1. Longkun Guo
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  2. Hong Shen
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Corresponding author

Correspondence to Hong Shen.

Additional information

This work is supported by National Science Foundation of China under its General Projects funding #61170232 and Youth funding #61100218, Fundamental Research Funds for the Central Universities #2012JBZ017, State Key Laboratory of Rail Traffic Control and Safety Research Grant RCS2011ZT009, and Research Initiative Grant of Beijing Jiaotong University # 2010RC018.

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Guo, L., Shen, H. On Finding Min-Min Disjoint Paths. Algorithmica 66, 641–653 (2013). https://doi.org/10.1007/s00453-012-9656-0

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