Hardness, approximability, and fixed-parameter tractability of the clustered shortest-path tree problem

Mattia D’Emidio (), Luca Forlizzi (), Daniele Frigioni (), Stefano Leucci () and Guido Proietti ()
Additional contact information
Mattia D’Emidio: University of L’Aquila
Luca Forlizzi: University of L’Aquila
Daniele Frigioni: University of L’Aquila
Stefano Leucci: ETH Zürich
Guido Proietti: University of L’Aquila

Journal of Combinatorial Optimization, 2019, vol. 38, issue 1, No 9, 165-184

Abstract: Abstract Given an n-vertex non-negatively real-weighted graph G, whose vertices are partitioned into a set of k clusters, a clustered network design problem on G consists of solving a given network design optimization problem on G, subject to some additional constraints on its clusters. In particular, we focus on the classic problem of designing a single-source shortest-path tree, and we analyse its computational hardness when in a feasible solution each cluster is required to form a subtree. We first study the unweighted case, and prove that the problem is $${\textsf {NP}}$$ NP -hard. However, on the positive side, we show the existence of an approximation algorithm whose quality essentially depends on few parameters, but which remarkably is an O(1)-approximation when the largest out of all the diameters of the clusters is either O(1) or $$\varTheta (n)$$ Θ ( n ) . Furthermore, we also show that the problem is fixed-parameter tractable with respect to k or to the number of vertices that belong to clusters of size at least 2. Then, we focus on the weighted case, and show that the problem can be approximated within a tight factor of O(n), and that it is fixed-parameter tractable as well. Finally, we analyse the unweighted single-pair shortest path problem, and we show it is hard to approximate within a (tight) factor of $$n^{1-\epsilon }$$ n 1 - ϵ , for any $$\epsilon >0$$ ϵ > 0 .

Keywords: Clustered shortest-path tree problem; Hardness; Approximation algorithms; Fixed-parameter tractability; Network design (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10878-018-00374-x 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:38:y:2019:i:1:d:10.1007_s10878-018-00374-x

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

DOI: 10.1007/s10878-018-00374-x

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 ().