On the union of intermediate nodes of shortest paths

Xiang Li; Xiaodong Hu; Wonjun Lee

doi:10.1007/s10878-011-9436-9

On the union of intermediate nodes of shortest paths

Xiang Li, Xiaodong Hu, Wonjun Lee

School of Cybersecurity

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

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,vεVM_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
Pages (from-to)	82-85
Number of pages	4
Journal	Journal of Combinatorial Optimization
Volume	26
Issue number	1
DOIs	https://doi.org/10.1007/s10878-011-9436-9
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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10.1007/s10878-011-9436-9

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title = "On the union of intermediate nodes of shortest paths",

abstract = "Consider a connected graph G=(V,E). For a pair of nodes u and v, denote by Muv the set of intermediate nodes of a shortest path between u and v. We are intertested in minimization of the union ∪ u,vεVMuv . 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.",

keywords = "Greedy approximation, Intersection of shortest paths",

author = "Xiang Li and Xiaodong Hu and Wonjun Lee",

note = "Funding Information: Acknowledgements This research was jointly sponsored by MEST, Korea under WCU (R33-2008-000-10044-0), MEST, Korea under Basic Science Research Program (2011-0012216), and MKE, Korea under ITRC NIPA-2011-(C1090-1121-0008).",

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AU - Hu, Xiaodong

AU - Lee, Wonjun

N1 - Funding Information: Acknowledgements This research was jointly sponsored by MEST, Korea under WCU (R33-2008-000-10044-0), MEST, Korea under Basic Science Research Program (2011-0012216), and MKE, Korea under ITRC NIPA-2011-(C1090-1121-0008).

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N2 - Consider a connected graph G=(V,E). For a pair of nodes u and v, denote by Muv the set of intermediate nodes of a shortest path between u and v. We are intertested in minimization of the union ∪ u,vεVMuv . 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.

AB - Consider a connected graph G=(V,E). For a pair of nodes u and v, denote by Muv the set of intermediate nodes of a shortest path between u and v. We are intertested in minimization of the union ∪ u,vεVMuv . 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.

KW - Greedy approximation

KW - Intersection of shortest paths

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On the union of intermediate nodes of shortest paths

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Keywords

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