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

Graduate School of Information Study

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

Access to Document

10.1007/s10878-011-9436-9

Cite this

@article{205a51de37f24ff590f6525aec2235e7,

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).",

year = "2013",

month = jul,

doi = "10.1007/s10878-011-9436-9",

language = "English",

volume = "26",

pages = "82--85",

journal = "Journal of Combinatorial Optimization",

issn = "1382-6905",

publisher = "Springer Netherlands",

number = "1",

}

TY - JOUR

T1 - On the union of intermediate nodes of shortest paths

AU - Li, Xiang

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).

PY - 2013/7

Y1 - 2013/7

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

UR - http://www.scopus.com/inward/record.url?scp=84878356043&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878356043&partnerID=8YFLogxK

U2 - 10.1007/s10878-011-9436-9

DO - 10.1007/s10878-011-9436-9

M3 - Article

AN - SCOPUS:84878356043

VL - 26

SP - 82

EP - 85

JO - Journal of Combinatorial Optimization

JF - Journal of Combinatorial Optimization

SN - 1382-6905

IS - 1

ER -

On the union of intermediate nodes of shortest paths

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this