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On the Terminal Steiner Tree Problem Information Processing Letters 84 (2002) 103–107 www.elsevier.com/locate/ipl On the terminal Steiner tree problem Guohui Lin a,1, Guoliang Xue b,∗,2 a Department of Computing Science, University of Alberta, Edmonton, AB, T6G 2E8, Canada b Department of Computer Science and Engineering, Arizona State University, Tempe, AZ 85287-5406, USA Received 29 October 2001; received in revised form 28 December 2001 Communicated by F.Y.L. Chin Abstract We investigate a practical variant of the well-known graph Steiner tree problem. In this variant, every target vertex is required to be a leaf vertex in the solution Steiner tree. We present hardness results for this variant as well as a polynomial time approximation algorithm with performance ratio ρ + 2, where ρ is the best-known approximation ratio for the graph Steiner tree problem. 2002 Elsevier Science B.V. All rights reserved. Keywords: Approximation algorithms; Steiner minimum tree; Terminal Steiner tree 1. Introduction non-target vertices for the sake of reducing weight. In the well-known graph Steiner tree problem These non-target vertices are called Steiner vertices (STP), we are given a positively edge-weighted graph and the tree achieving the minimum weight is called and a subset of target vertices in the graph. The goal a Steiner minimum tree (SMT) for the target vertices. is to find a minimum weight subgraph interconnecting The STP, as well as the MST problem, has all target vertices, where the weight of the subgraph applications in many practical problems such as VLSI is the sum of the weights of the edges therein. From global and local routing, telecommunications, and the nature of the problem, it is assumed that the transportation. It has received a lot of attention in the given graph is connected. The minimization objective last several decades and some of the important results implies that the interconnection subgraph is a tree. To can be found in [9,3,2,6,12,1,5,4,8,11], which include distinguish from the minimum spanning tree (MST) hardness results, (in)approximabilities, and quite a few problem, the tree computed for STP may contain some approximation algorithms (see also [7] for a survey) together with their performance analyses. * Corresponding author. In some applications, the target vertices are re- E-mail addresses: [email protected] (G. Lin), quired to be leaves in the Steiner tree. For example, in [email protected] (G. Xue). 1 VLSI global routing, the target vertices correspond to Research supported in part by Startup grant G227120195 from pins and gates that are not allowed to overlap with the the University of Alberta. 2 Research supported in part by ARO grant DAAD19-00-1-0377 Steiner vertices introduced to reduce the total weight. and DOE grant DE-FG02-00ER45828. In telecommunications, the message senders and re- 0020-0190/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII:S0020-0190(02)00227-2 104 G. Lin, G. Xue / Information Processing Letters 84 (2002) 103–107 ceivers are not allowed to act as transmitters which from STP on graphs whose edge weights are either 1 correspond to the Steiner vertices. Therefore, in the so- or 2. A graph whose edge weights are either 1 or 2 is lution tree, it is desirable to have all target vertices as called a 1, 2-edge-weighted graph. leaf vertices. We study the following terminal Steiner tree problem or TEST, for short, which is a graph STP Lemma 2.1 [3]. STP restricted to the 1, 2-edge- problem with an additional requirement that every tar- weighted graphs is MAX SNP-hard. get vertex must be a leaf vertex in the output intercon- nection tree. Given a 1, 2-edge-weighted complete graph G with For feasibility, we assume that there exists an vertex set U and target set S ⊂ U, for each vertex a ∈ optimal solution tree in which every target vertex S create a new vertex t(a) and a new edge (a, t(a)) appears as a leaf vertex. Therefore, in the given where t(a) is taken as a target vertex in the TEST graph every target vertex should be adjacent to some instance I.ThetargetsetinI is T ={t(a), a∈ S} and non-target vertex. Furthermore, we may compute, the vertex set is V = U ∪ T . Note that a is no longer for each pair of vertices u and v, a shortest u–v a target in instance I. The weight of edge (u, v) for path (with weight denoted by w(u,v)) that does not u, v ∈ U is not changed. The weight of edge (a, t(a)) contain any target vertex as an internal vertex. We is set at ε, which is a sufficiently small positive number then add the edge (u, v) (or substitute if the edge less than 1/|S|. Completing the above graph gives the is already in) with weight w(u,v) into the graph. instance I of TEST, denoted by N(V,w,T ). This operation will not change the corresponding Based on the above polynomial time transforma- Steiner tree problem. As a result, we may assume tion, it is easy to see that whenever there is a Steiner without loss of generality that the given graph is tree S interconnecting vertices in S in graph G,we complete and the edge weights satisfy the triangle may trivially construct a Steiner tree T interconnect- inequality, that is, for every three vertices u, v, x: ing target vertices in T in graph N by adding edges w(u,v) w(u,x) + w(x,v). Throughout the paper, in the form (a, t(a)) for all a ∈ S into S. One can an approximation algorithm means a polynomial time verify that w(T ) = c(S) +|S|ε<c(S)+ 1. On the approximation algorithm. We use ρ to denote the best- other hand, given any feasible Steiner tree T inter- known approximation ratio for the STP. 1 connecting target vertices in T in graph N (every tar- Let N(V,w,T ) be a positively edge-weighted get appears as a leaf in T ), we can get another feasible complete graph with vertex set V , edge weighting 1 Steiner tree T , with weight no more than w(T ),inter- function w satisfying the triangle inequality, and target 1 connecting targets in T in which every target t(a)∈ T set T ⊂ V .Theterminal Steiner tree problem (TEST) is connected to vertex a. Deleting |T | edges of form asks for a minimum weight subgraph interconnecting (a, t(a)) for all a ∈ S, we will get a tree S intercon- all target vertices such that every target vertex appears necting vertex in S in graph N,andS does not contain as a leaf vertex in the subgraph. any edges outside G. Thus, S is a Steiner tree inter- In Section 2, we will show that the TEST problem connecting vertices in S in graph G. Furthermore, the is NP-hard and MAX SNP-hard (APX-hard) [10]. In weight of tree S is less than the weight of T . Therefore Section 3, we will present a polynomial time approxi- we have the following hardness result. mation algorithm for TEST with a approximation ratio ρ + 2, where ρ is the best-known approximation ratio for STP. We conclude this paper in Section 4. Theorem 2.2. The TEST problem is MAX SNP-hard. 2. Hardness 3. Approximations In this section, we will show that the TEST problem In this section, we will present two provably good is MAX SNP-hard. Because the techniques used here polynomial time approximation algorithms for TEST. are standard reductions, we will present a high level The first approximation algorithm is easy to under- description only. The proof is done via an L-reduction stand, with an approximation ratio of 1 + 2ρ. The sec- G. Lin, G. Xue / Information Processing Letters 84 (2002) 103–107 105 ALGORITHM I shrink phase: for each target t ∈ T ,findvertexs(t); frame phase: compute a ρ-approximation T0 to the optimal Steiner tree interconnecting {s(t),t ∈ T } in the induced subgraph N[V − T ]; assembly phase: add to T0 edges of form (t, s(t)),forallt ∈ T , to obtain T ; output the resulting tree T . Fig. 1. A high-level description of the first approximation. ond approximation algorithm is more involved, with vertices in {s(t),t ∈ T } in the induced subgraph an improved approximation ratio of 2 + ρ. N[V −T ]. Denote this Steiner tree using T . It follows Our first approximation algorithm consists of three from triangle inequality w(s∗(t), s(t)) w(s∗(t), t)+ phases: shrink, frame,andassembly.LetN(V,w,T ) w(t,s(t)) that be a given instance of the TEST problem. In the ∗ + shrink phase, we search for each target t ∈ T a w(T ) w(T ) w s(t),t . (3.3) Steiner vertex s(t) ∈ V − T such that w(t,s(t)) is t∈T the minimum over all choices of Steiner vertices (ties Let T ∗ denote a minimum weight Steiner tree in- are broken arbitrarily). In the frame phase, we restrict terconnecting vertices in {s(t),t ∈ T } in the induced our attention to computing an approximate Steiner tree subgraph N[V − T ] and recall that T0 denotes the in the induced subgraph N[V − T ] where the set of Steiner tree that we compute to interconnect vertices “target” vertices is {s(t), t ∈ T }. We can employ the in {s(t),t ∈ T } in the induced subgraph N[V − T ].
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