### Abstract

Consider a connected graph G=(V,E). For a pair of nodes u and v, denote by M_{uv} the set of intermediate nodes of a shortest path between u and v. We are intertested in minimization of the union ∪ _{u,vεV}M_{uv} . We will show that this problem is NP-hard and cannot have polynomial-time ρlnδ-approximation for 0<ρ<1 unless NP ⊆ DTIME(n ^{O(loglogn)}) where δ is the maximum node degree of input graph. We will also construct a polynomial-time H(Δ(Δ-1)/2) -approximation for the problem where H(·) is the harmonic function.

Original language | English |
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Pages (from-to) | 82-85 |

Number of pages | 4 |

Journal | Journal of Combinatorial Optimization |

Volume | 26 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2013 Jul |

### Keywords

- Greedy approximation
- Intersection of shortest paths

### ASJC Scopus subject areas

- Computer Science Applications
- Discrete Mathematics and Combinatorics
- Control and Optimization
- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

Li, X., Hu, X., & Lee, W. (2013). On the union of intermediate nodes of shortest paths.

*Journal of Combinatorial Optimization*,*26*(1), 82-85. https://doi.org/10.1007/s10878-011-9436-9