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

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## 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 \({\textsf {NP}}\)-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 \(\epsilon >0\).

## Keywords

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

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