The internal Steiner tree problem: Hardness and approximations

Huang, Chun-Rong; Lee, Chia‐Wei; Gao, Huang-Ming; Hsieh, Sun-Yuan

doi:10.1016/j.jco.2012.08.005

Cited by 11 publications

(5 citation statements)

References 27 publications

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“…The current best approximation ratio is 1.39 (Byrka et al 2010). Numerous variants of the SMT problem have been studied, for example, the versions on the Euclidean metric ) and the rectilinear metric , the Steiner forest problem (Agrawal et al 1995), the group Steiner tree problem Garg et al (1998), the terminal Steiner tree problem (Chen et al 2003;Drake 2004;Lin and Xue 2002;Lu et al 2003;Martinez et al 2007), the internal-selected Steiner tree problem (Hsieh and Yang 2007;Huang et al 2013;Li et al 2010), and many others (Ding and Xue 2014;Hsu et al 2005;Zou et al 2009). …”

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confidence: 99%

On the clustered Steiner tree problem

Lin

2014

J Comb Optim

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We investigate the Clustered Steiner tree problem on metric graphs, which is a variant of the Steiner minimum tree problem. In this problem, the required vertices are partitioned into clusters, and the subtrees spanning different clusters must be disjoint in a feasible clustered Steiner tree. In this paper, it is shown that the problem is NPhard even if the inter-cluster tree and all the local topologies are given, where a local topology specifies the tree structure of required vertices in the same cluster. We show that the Steiner ratio of this problem is lower and upper bounded by three and four, respectively. We also propose a (ρ + 2)-approximation algorithm, where ρ is the approximation ratio for the Steiner minimum tree problem, and the approximation ratio can be improved to ρ + 1 if the local topologies are given. Two variants of this problem are also studied. When the goal is to minimize the inter-cluster cost and ignore the cost of local trees, the problem can be solved in polynomial time. But it is NP-hard if we ask for the minimum cost of local trees among all solutions with minimum inter-cluster cost.

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mentioning

confidence: 99%

On the clustered Steiner tree problem

Lin

2014

J Comb Optim

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“…• Is Internal Steiner Tree(k + |T |) [24] diminishable? • Clique(∆), Clique(tw), Clique(bw) do not have strong diminishers under the ETH (Section 4).…”

Section: Discussionmentioning

confidence: 99%

Diminishable parameterized problems and strict polynomial kernelization

Fernau

Fluschnik

Hermelin

et al. 2020

COM

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Kernelization-a mathematical key concept for provably effective polynomial-time preprocessing of NP-hard problems-plays a central role in parameterized complexity and has triggered an extensive line of research. This is in part due to a lower bounds framework that allows to exclude polynomial-size kernels under the assumption of NP coNP/poly. In this paper we consider a restricted yet natural variant of kernelization, namely strict kernelization, where one is not allowed to increase the parameter of the reduced instance (the kernel) by more than an additive constant.Building on earlier work of Chen, Flum, and Müller [Theory Comput. Syst. 2011] and developing a general and remarkably simple framework, we show that a variety of FPT problems does not admit strict polynomial kernels under the weaker assumption of P = NP. In particular, we show that various (multicolored) graph problems and Turing machine computation problems do not admit strict polynomial kernels unless P = NP. To this end, a key concept we use are diminishable problems; these are parameterized problems that allow to decrease the parameter of the input instance by at least one in polynomial time, thereby outputting an equivalent problem instance. Finally, we study a relaxation of the notion of strict kernels and reveal its limitations.

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“…Given a graph G and a set of terminal vertices S ⊆ V(G), the goal is to identify as many edgedisjoint S-Steiner trees (i.e., trees T in G with S ⊆ V(T)) as feasible. This particular problem, along with its associated topics, garners significant interest from researchers due to its extensive applications in VLSI circuit design [2][3][4] and Internet Domain [5]. In practical applications, the construction of vertex-disjoint or arc-disjoint paths in graphs holds significance, as they play a crucial role in improving transmission reliability and boosting network transmission rates [6].…”

Section: Introductionmentioning

confidence: 99%

Directed Path 3-Arc-Connectivity of Cartesian Product Digraphs

Wei

2024

Symmetry

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Let D=(V(D),A(D)) be a digraph of order n and let r∈S⊆V(D) with 2≤|S|≤n. A directed (S,r)-Steiner path (or an (S,r)-path for short) is a directed path P beginning at r such that S⊆V(P). Arc-disjoint between two (S,r)-paths is characterized by the absence of common arcs. Let λS,rp(D) be the maximum number of arc-disjoint (S,r)-paths in D. The directed path k-arc-connectivity of D is defined as λkp(D)=min{λS,rp(D)∣S⊆V(D),S=k,r∈S}. In this paper, we shall investigate the directed path 3-arc-connectivity of Cartesian product λ3p(G□H) and prove that if G and H are two digraphs such that δ0(G)≥4, δ0(H)≥4, and κ(G)≥2, κ(H)≥2, then λ3p(G□H)≥min2κ(G),2κ(H); moreover, this bound is sharp. We also obtain exact values for λ3p(G□H) for some digraph classes G and H, and most of these digraphs are symmetric.

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The internal Steiner tree problem: Hardness and approximations

Cited by 11 publications

References 27 publications

On the clustered Steiner tree problem

On the clustered Steiner tree problem

Diminishable parameterized problems and strict polynomial kernelization

Directed Path 3-Arc-Connectivity of Cartesian Product Digraphs

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