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

  • Mattia D’EmidioEmail author
  • Luca Forlizzi
  • Daniele Frigioni
  • Stefano Leucci
  • Guido Proietti


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 \({\textsf {NP}}\)-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 \(\varTheta (n)\). 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 \(n^{1-\epsilon }\), for any \(\epsilon >0\).


Clustered shortest-path tree problem Hardness Approximation algorithms Fixed-parameter tractability Network design 



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Authors and Affiliations

  1. 1.Department of Information Engineering, Computer Science and MathematicsUniversity of L’AquilaL’AquilaItaly
  2. 2.Department of Computer ScienceETH ZürichZürichSwitzerland
  3. 3.Istituto di Analisi dei Sistemi e Informatica “Antonio Ruberti” Consiglio Nazionale delle RicercheRomeItaly

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