Hardness, Approximability, and Fixed-Parameter Tractability of the Clustered Shortest-Path Tree Problem

• Published in: arXiv.org
• Main Authors: D'Emidio, Mattia, lizzi, Luca, Frigioni, Daniele, Leucci, Stefano, Proietti, Guido
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 31.01.2018
• Subjects: Algorithms; Apexes; Approximation; Clusters; Design optimization; Feasibility studies; Graph theory; Mathematical analysis; Parameters; Shortest-path problems
• ISSN: 2331-8422

Given an \(n\)-vertex non-negatively real-weighted graph \(G\), whose vertices are partitioned into a set of \(k\) clusters, a \emph{clustered network design problem} on \(G\) consists of solving a given network design optimization problem on \(G\), subject to some additional constraint on its clusters. In particular, we focus on the classic problem of designing a \emph{single-source shortest-path tree}, and we analyze its computational hardness when in a feasible solution each cluster is required to form a subtree. We first study the \emph{unweighted} case, and prove that the problem is \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 \emph{diameters} of the clusters is either \(O(1)\) or \(\Theta(n)\). Furthermore, we also show that the problem is \emph{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 \emph{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 analyze the unweighted \emph{single-pair shortest path problem}, and we show it is hard to approximate within a (tight) factor of \(n^{1-\epsilon}\), for any \(\epsilon>0\).