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Parameterized complexity of finding a spanning tree with minimum reload cost diameter
Author(s) -
Baste Julien,
Gözüpek Didem,
Paul Christophe,
Sau Ignasi,
Shalom Mordechai,
Thilikos Dimitrios M.
Publication year - 2020
Publication title -
networks
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.977
H-Index - 64
eISSN - 1097-0037
pISSN - 0028-3045
DOI - 10.1002/net.21923
Subject(s) - treewidth , parameterized complexity , mathematics , combinatorics , spanning tree , bounded function , tree (set theory) , steiner tree problem , time complexity , graph , path (computing) , minimum spanning tree , gomory–hu tree , k minimum spanning tree , discrete mathematics , pathwidth , tree structure , k ary tree , binary tree , computer science , line graph , mathematical analysis , programming language
We 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.