Parameterized complexity of finding a spanning tree with minimum reload cost diameter
Julien Baste, Didem Gözüpek, Christophe Paul, Ignasi Sau, Mordechai Shalom, Dimitrios M. Thilikos · Networks · 2019
Abstract We study the minimum diameter spanning tree problem under the reload cost model (Diameter‐Treefor short) introduced by Wirth and Steffan. In this problem, given an undirected edge‐colored graphG, 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 ofGof minimum diameter with respect to the reload costs. We initiate a systematic study of the parameterized complexity of theDiameter‐Treeproblem by considering the following parameters: the cost of a solution, and the treewidth and the maximum degree Δ of the input graph. We prove thatDiameter‐Treeispara‐NP‐hard for any combination of two of these three parameters, and that it isFPTparameterized 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 proveDiameter‐Treeto beNP‐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 isNP‐hard if Δ = 4. Our results show, in particular, that without the requirement of the triangle inequality, the problem isNP‐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 thatDiameter‐Treeis inXPandW[1]‐hard parameterized by the treewidth plus Δ.