An FPT algorithm for node-disjoint subtrees problems parameterized by treewidth

In this paper, we introduce a problem called Minimum subTree problem with Degree Weights, or MTDW. This problem generalized covering tree problems like Spanning Tree, Steiner Tree, Minimum Branch Vertices, Minimum Leaf Spanning Tree, or Prize Collecting Steiner Tree. It consists, given an undirected...

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Vydáno v:Theoretical computer science Ročník 990; s. 114406
Hlavní autoři: Baste, Julien, Watel, Dimitri
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier B.V 01.04.2024
Elsevier
Témata:
Computer Science
Fixed-parameter tractable algorithm
Graph algorithm
Spanning tree problems
Treewidth
Spanning tree problems
Fixed-parameter tractable algorithm
Graph algorithm
Treewidth
ISSN:0304-3975, 1879-2294
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Shrnutí:In this paper, we introduce a problem called Minimum subTree problem with Degree Weights, or MTDW. This problem generalized covering tree problems like Spanning Tree, Steiner Tree, Minimum Branch Vertices, Minimum Leaf Spanning Tree, or Prize Collecting Steiner Tree. It consists, given an undirected graph G=(V,E), a set of m+1 mappings C1,C2,…,Cm,D:V×N→Z, a set of m integers K1,K2,…,Km∈Z and a positive integer ℓ, in the search of a forest (T1,T2,…,Tℓ) containing ℓ node-disjoint trees of G. Along with Kj, the mapping Cj defines a constraint that should be satisfied by the trees of the forest. For each tree Ti, it associates each node v of V to the score Cj(v,dTi(v)) where dTi(v) is the degree of v in Ti (possibly 0 if the node is not in Ti). The sum ∑v∈VCj(v,dTi(v)) should not exceed Kj. In addition, the forest should minimize ∑i=1ℓ∑v∈VD(v,dTi(v)). We proceed to a parameterized analysis of the MTDW problem with regard to four parameters that are the number of constraints m, the value ℓ, the treewidth of the input graph G and Δ, the minimum degree above which all the constraints and D are constant (for every j∈〚1,m〛, v∈V and d≥Δ, Cj(v,d)=Cj(v,Δ), and D(v,d)=D(v,Δ)). For this problem, we provide a first dichotomy P versus NP-hard depending whether the previous parameters are fixed to be constant or not and a second dichotomy FPT versus W[1]-hard depending whether each of these parameters is constant, considered as a parameter, or disregard. As a side effect, we obtained parameterized algorithms, previously undescribed, for problems such that Budget Steiner Tree problem with Profits, Minimum Branch Vertices, Generalized branch vertices, or k-Bottleneck Steiner Tree.
ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2024.114406