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

• Published in: Journal of combinatorial optimization Vol. 38; no. 1; pp. 165 - 184
• Main Authors: D’Emidio, Mattia, Forlizzi, Luca, Frigioni, Daniele, Leucci, Stefano, Proietti, Guido
• Format: Journal Article
• Language: English
• Published: New York Springer US 15.07.2019; Springer Nature B.V
• Subjects: Algorithms; Apexes; Approximation; Clusters; Combinatorics; Convex and Discrete Geometry; Design optimization; Feasibility studies; Graph theory; Mathematical analysis; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Parameters; Shortest-path problems; Theory of Computation; Approximation algorithms; Network design; Hardness; Clustered shortest-path tree problem; Fixed-parameter tractability
• ISSN: 1382-6905; 1573-2886
• DOI: 10.1007/s10878-018-00374-x

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 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 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 - ϵ , for any ϵ > 0 .