Abstract
AbstractWe study the minimum diameter spanning tree problem under the reload cost model (Diameter‐Tree for short) introduced by Wirth and Steffan. In this problem, given an undirected edge‐colored graph G, reload costs on a path arise at a node where the path uses consecutive edges of different colors. The objective is to find a spanning tree of G of minimum diameter with respect to the reload costs. We initiate a systematic study of the parameterized complexity of the Diameter‐Tree problem by considering the following parameters: the cost of a solution, and the treewidth and the maximum degree Δ of the input graph. We prove that Diameter‐Tree is para‐NP‐hard for any combination of two of these three parameters, and that it is FPT parameterized by the three of them. We also prove that the problem can be solved in polynomial time on cactus graphs. This result is somehow surprising since we prove Diameter‐Tree to be NP‐hard on graphs of treewidth two, which is best possible as the problem can be trivially solved on forests. When the reload costs satisfy the triangle inequality, Wirth and Steffan proved that the problem can be solved in polynomial time on graphs with Δ = 3, and Galbiati proved that it is NP‐hard if Δ = 4. Our results show, in particular, that without the requirement of the triangle inequality, the problem is NP‐hard if Δ = 3, which is also best possible. Finally, in the case where the reload costs are polynomially bounded by the size of the input graph, we prove that Diameter‐Tree is in XP and W[1]‐hard parameterized by the treewidth plus Δ.
Highlights
Numerous network optimization problems can be modeled by edge-colored graphs
Wirth and Steffan introduced in [28] the concept of reload cost, which refers to the cost that arises in an edge-colored graph while traversing a vertex via two consecutive edges of different colors
Many works in the cognitive radio networks (CRN) literature focused on this frequency switching cost from an application point of view by analyzing its various aspects such as delay and energy consumption
Summary
Numerous network optimization problems can be modeled by edge-colored graphs. Wirth and Steffan introduced in [28] the concept of reload cost, which refers to the cost that arises in an edge-colored graph while traversing a vertex via two consecutive edges of different colors. We prove, by a reduction from a restricted version of 3-Sat, that Diameter-Tree is NP-hard on graphs with ∆ ≤ 3, even with only two different costs, k = 0, and a bounded number of colors This settles the complexity of the problem on graphs with ∆ ≤ 3 in the general case where the triangle inequality is not necessarily satisfied, which had been left open in previous work [28, 12]. Due to lack of space, the proof of the results marked with ‘[ ]’ can be found in the full version
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