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 \(\epsilon >0\).
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Notes
Throughout the paper, the notation \(\widetilde{O}\) suppresses factors that are polylogarithmic in n.
The runtime originally given in Björklund et al. (2007) is here restated on our (implicitly assumed) model of computation, namely the standard unit-cost RAM with logarithmic word size, on which the \(O(|X|^2 \cdot 2^{|X|})\) ring operations performed in Björklund et al. (2007) cost \(O(W \cdot |X| \cdot {\text {polylog}}(W,|X|))\) time each. Notice that we are explicitly stating polynomial factors in |X|, i.e., logarithmic factors in \(2^{|X|}\), which are disregarded in Björklund et al. (2007), since they will result in polynomial factors in k in the running time of our FPT algorithm.
The X3C problem remains NP-complete even with this additional assumption, see e.g., problem SP2 in Garey and Johnson (1979).
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The results presented in this work have been announced in a preliminary form in D’Emidio et al. (2016). This research has partially supported by the Italian National Group for Scientific Computation GNCS-INdAM.
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D’Emidio, M., Forlizzi, L., Frigioni, D. et al. Hardness, approximability, and fixed-parameter tractability of the clustered shortest-path tree problem. J Comb Optim 38, 165–184 (2019). https://doi.org/10.1007/s10878-018-00374-x
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DOI: https://doi.org/10.1007/s10878-018-00374-x