EconPapers    
Economics at your fingertips  
 

W[1]-hardness of the k-center problem parameterized by the skeleton dimension

Johannes Blum ()
Additional contact information
Johannes Blum: Universität Konstanz, Universitätsstr. 10

Journal of Combinatorial Optimization, 2022, vol. 44, issue 4, No 32, 2762-2781

Abstract: Abstract We study the k -Center problem, where the input is a graph $$G=(V,E)$$ G = ( V , E ) with positive edge weights and an integer k, and the goal is to select k center vertices $$C \subseteq V$$ C ⊆ V such that the maximum distance from any vertex to the closest center vertex is minimized. In general, this problem is $$\mathsf {NP}$$ NP -hard and cannot be approximated within a factor less than 2. Typical applications of the k -Center problem can be found in logistics or urban planning and hence, it is natural to study the problem on transportation networks. Common characterizations of such networks are graphs that are (almost) planar or have low doubling dimension, highway dimension or skeleton dimension. It was shown by Feldmann and Marx that k -Center is $$\mathsf {W[1]}$$ W [ 1 ] -hard on planar graphs of constant doubling dimension when parameterized by the number of centers k, the highway dimension $$hd$$ hd and the pathwidth $$pw$$ pw (Feldmann and Marx 2020). We extend their result and show that even if we additionally parameterize by the skeleton dimension $$\kappa $$ κ , the k -Center problem remains $$\mathsf {W[1]}$$ W [ 1 ] -hard. Moreover, we prove that under the Exponential Time Hypothesis there is no exact algorithm for k -Center that has runtime $$f(k,hd,pw,\kappa ) \cdot \vert V \vert ^{o(pw+ \kappa + \sqrt{k+hd})}$$ f ( k , h d , p w , κ ) · | V | o ( p w + κ + k + h d ) for any computable function f.

Keywords: k-Center; Skeleton dimension; Highway dimension; Parameterized complexity (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10878-021-00792-4 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jcomop:v:44:y:2022:i:4:d:10.1007_s10878-021-00792-4

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10878

DOI: 10.1007/s10878-021-00792-4

Access Statistics for this article

Journal of Combinatorial Optimization is currently edited by Thai, My T.

More articles in Journal of Combinatorial Optimization from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().

 
Page updated 2025-03-20
Handle: RePEc:spr:jcomop:v:44:y:2022:i:4:d:10.1007_s10878-021-00792-4