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Two-connected Steiner networks: structural properties
Operations Research Letters 33 (2005) 395 – 402
Operations Research Letters www.elsevier.com/locate/dsw
Two-connected Steiner networks: structural properties Pawel Winter, Martin Zachariasen∗ Department of Computer Science, University of Copenhagen, Universitetsparken 1, DK-2100 Copenhagen ], Denmark Received 9 January 2004; accepted 29 July 2004 Available online 6 October 2004
Abstract We give a number of structural results for the problem of constructing a minimum-length 2-connected network for a set of terminals in a graph, where edge-weights satisfy the triangle inequality. A new algorithmic framework, based on our structural results, is given. © 2004 Elsevier B.V. All rights reserved. Keywords: Survivable networks; 2-connected Steiner networks
1. Introduction The well-known Steiner tree problem asks for a shortest possible network spanning a set Z of terminals in the plane. The solution to the problem is a tree, referred to as a Steiner minimal tree (SMT). Apart from the terminals that must be spanned, the SMT may contain additional, so-called Steiner points, where exactly three edges meet at 120◦ angles. SMTs are unions of full Steiner trees spanning subsets of terminals all having degree 1. When the objective is to design low-cost survivable networks, the problem of constructing 2-connected Steiner minimal networks in the plane arises. In a 2-edge-connected (resp. 2-vertex connected) Steiner ∗ Corresponding author.
E-mail addresses: [email protected] (P. Winter), [email protected] (M. Zachariasen). 0167-6377/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2004.07.010
network there are at least two edge-disjoint (resp. vertex-disjoint) paths between every pair of terminals. In the following, we do not distinguish between the vertex and edge connected variants of the problem for reasons to be explained in Section 2. The 2-connected Steiner network problem in the plane has been studied by Luebke and Provan [6], who proved that it is NP-hard and gave a number of structural properties. Additional structural properties for the generalized graph version where distances fulfill the triangle inequality were given by Luebke [5]. In this paper, we significantly improve and generalize the results given in [5,6]. We show that all cycles must have at least four terminals, and that each cycle must contain at least two non-incident edges connecting terminals. This allows us to give a tight lower bound on the number of terminals needed in any instance that requires a Steiner vertex in a minimumlength solution.
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The paper is organized as follows. In Section 2 we formally define the problems that are considered; we also review some of the known results. In Section 3 we prove some properties of the so-called chord-paths, and in Section 4 we give our main results on the distribution of terminals in cycles of 2-connected Steiner networks. Lower bounds (and tight examples) on the size of Steiner networks that require Steiner vertices are given in Section 5. In Section 6 we present an algorithmic framework, based on the structural results given in this paper, for solving the problem. Preliminary computational results and concluding remarks are given in Section 7.
2. Preliminaries Let G = (V , E) be a complete undirected graph with a distance function defined on its vertices. For a pair of vertices u, v ∈ V , let d(u, v) denote the distance between u and v. The distance function is assumed to fulfill the requirements of a metric, i.e., it is non-negative, symmetric and satisfies the triangle inequality. Finally, let Z ⊆ V be a set of terminals; the remaining vertices V \Z are called Steiner vertices. The 2-connected Steiner network problem (2SNPG) is to find subgraph G = (V , E ) of G such that • G is 2-edge-connected, • Z ⊆ V , • G has the minimum total length. Since a 2-edge-connected minimum-length network necessarily is 2-vertex-connected when the distance function is a metric [7], we use the shorthand 2connected in the following. This problem was studied by Monma et al. [7], where the main focus was on the special case when the graph contains no Steiner vertices, that is, when Z = V . They proved that there always exists an optimal solution in which all terminals have degree 2 or 3, and all Steiner vertices have degree 3. Hsu and Hu [3] and Luebke and Provan [6] considered a special case of the graph problem, namely the Euclidean 2-connected Steiner network problem in the plane (2-SNPP). In this problem, the terminals Z are points in the plane, and the task is to construct
a minimum-length 2-connected network that interconnects Z. (The problem is a special case of the graph problem since only a finite set of Steiner vertices needs to be considered due to the 120◦ angle conditions at Steiner vertices.) For this problem it was proved that Steiner vertices are incident to three edges meeting at 120◦ angles (as for the Euclidean Steiner tree problem in the plane), and that no cycles in a minimum-length network consist entirely of Steiner vertices. As a consequence, a shortest network is a union of full Steiner trees (FSTs), in which all terminals are leaves and all Steiner vertices are interior vertices. In this paper, we study the structure of minimumlength solutions for the general graph problem (2SNPG)—which contains 2-SNPP as a special case. More specifically, we will study the structure of minimum-length solutions in which the total degree of all vertices is minimized. In this way we avoid trivial degeneracies. We denote by SMN, an arbitrary minimum-length 2-connected network for which the total degree of all vertices has been minimized. (For 2-SNPP an arbitrary 2-connected minimum-length network is denoted by SMNP). Luebke [5] proved that the following properties hold for any SMN: • All vertices have degree 2 or 3, and all Steiner vertices have degree 3. • All edges are of multiplicity one (unless |Z| = 2). • No cycle is composed entirely of Steiner vertices. • For |Z| 4 there are no cycles with exactly three vertices.
3. Chord-path properties Let G denote an arbitrary undirected graph. Given a cycle C and two distinct vertices u and v on C in G, a chord-path between u and v is a path P (u, v) in G between u and v that, except from u and v, shares neither vertices nor edges with C. Note that the interior vertices in P (u, v) are not required to have degree 2 in G. When P (u, v) consists of a single edge, the chordpath reduces to a simple chord edge of C. It is well known [5,7] that an SMN cannot have any chord edge: Consider a cycle C having a chord edge (u, v). Clearly, the subgraph C ∪(u, v) is 2-connected. If we delete the edge (u, v) from this subgraph, the total length of the network does not increase and the
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(b) Fig. 1. (a and b) A chord-path consisting of two edges.
total degree decreases. Furthermore, since the remaining subgraph (which is C) is still 2-connected, the resulting overall network remains 2-connected [7]. As a consequence, any chord-path in an SMN must have at least two edges. The following strengthens this result. Lemma 1. Any chord-path in an SMN must have at least three edges. Proof. Assume that there exists a cycle C with a chord-path P (u, v) consisting of two edges (u, t) and (t, v). Let a and b be the neighbours of u on C, and let c and d be the neighbours of v on C (see Fig. 1(a)). Neither vertices a and d nor vertices b and c are required to be distinct. Assume w.l.o.g. that d(t, v) d(t, u). Remove the edges (t, u) and (v, c), and add the edge (t, c), as shown in Fig. 1(b). The network remains 2-connected and its length does not increase: d(t, c) d(t, v) + d(v, c) d(t, u) + d(v, c). Both vertices u and v have reduced their degree with one, while the degree of all other vertices is unchanged. Thus we have arrived at a contradiction to the (length and degree) minimality of SMN. Luebke and Provan [6] showed for 2-SNPP that an SMNP cannot have a chord-path that has only Steiner vertices in its interior. Using Lemma 1, we can now generalize this result. Theorem 1. Any chord-path in an SMN must have a pair of consecutive terminals of degree 2 in its interior. Proof. Consider a chord-path with no consecutive terminals of degree 2 in its interior; among all such chord-
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paths, let P (u, v) be one with the minimum number of edges, and let C be a cycle for which P (u, v) is a chord-path. By Lemma 1, P (u, v) has at least two interior vertices, and hence at least one interior vertex w of degree 3—otherwise P (u, v) would have two consecutive terminals of degree 2 in its interior. Let x be the third vertex adjacent to w and not on P (u, v) (Fig. 2a). Edge (w, x) must be on some cycle Cw in SMN [7]. The cycle Cw uses one of the edges incident to w on P (u, v), since w has degree 3. As a consequence, Cw will share at least one vertex with C ∪ P (u, v), and distinct from w. Follow the cycle Cw , starting in x and moving away from w. At some point the cycle Cw will encounter a vertex y = w on C ∪ P (u, v); note that vertex y is distinct from w as argued above. Assume first that y is some vertex on P (u, v). Then the subpath of P (u, v) from w to y is a chord-path of another cycle in SMN, as shown in Fig. 2b. This chordpath has fewer edges than P (u, v). Furthermore, it has no pair of consecutive degree 2 terminals in its interior. This is a contradiction to the choice of P (u, v). Now assume that y is a vertex on C, distinct from u and v. The subpath of P (u, v) from u to w is a chordpath of another cycle, as shown in Fig. 2c. Again, this chord-path has fewer edges than P (u, v) and has no pair of consecutive degree 2 terminals in its interior. Once again this is a contradiction to the choice of P (u, v). In conclusion, chord-path P (u, v) cannot exist.
4. Cycle properties In this section, we use the properties of chord-paths to prove a number of fundamental structural properties for cycles in solutions to 2-SNPG. Theorem 2. Consider a cycle C that contains a vertex of degree 3 in an SMN. Then C must have two pairs of consecutive terminals, both of degree 2 in SMN. Furthermore, these two terminal pairs must be separated on C by a pair vertices of degree 3. Proof. Let w be a vertex on C of degree 3. Let x be the vertex adjacent to w not on C. Edge (w, x) must be on some cycle Cw in SMN; this cycle uses one of
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Fig. 2. (a, b and c) A chord-path with a degree 3 vertex.
the edges on C incident to w (since w has degree 3). As a consequence, Cw will share at least one vertex with C, and is distinct from w. Follow the cycle Cw , starting in x and moving away from w. Let y be the first vertex on C that is encountered, and define P1 to be the path from w to y following the cycle Cw . Let P2 and P3 denote the two edge-disjoint paths from y to w on C. Note that the paths P1 , P2 and P3 share no vertices except from their endpoints. Consider the cycle P1 ∪ P2 . The path P3 is a chord-path of this cycle and by Theorem 1 it has a pair of consecutive terminals of degree 2 in SMN. Now consider the cycle P1 ∪ P3 . The path P2 is a chord-path of this cycle and must also have a pair of consecutive terminals of degree 2 in SMN. The theorem follows. Consider an instance of 2-SNPG with at least four terminals. Either the SMN for this problem is a simple cycle through all terminals, or every cycle has two vertices of degree 3. In both cases we get the following: Corollary 1. If the total number of terminals in an SMN is at least four, then every cycle in the SMN has at least four terminals of degree 2. An SMN is a union of full Steiner trees (FSTs) [6]. Consider a cycle C in an SMN in which every edge is from an FST spanning at least three terminals (Fig. 3). Thus every terminal in C has two Steiner vertices as neighbors, that is, cycle C has no pair of consecutive terminals of degree 2 as required by Theorem 2. We have the following corollary: Corollary 2. No cycle in an SMN has edges solely from full Steiner trees spanning three or more terminals.
Fig. 3. A cycle with edges solely from FSTs spanning three or more terminals. In this example, the cycle contains 6 terminals; the remaining vertices on the cycle are Steiner vertices in FSTs spanning three or more terminals.
5. Smallest networks with Steiner vertices In this section, we will show that an SMN cannot have vertices of degree 3 unless it contains at least 6 terminals. We will also show that this bound is tight for 2-SNPG. We conjecture that the smallest number of terminals needed for an SMNP (i.e., in the Euclidean plane) to have vertices of degree 3 is 8. We also give a problem instance with 8 terminals where the SMNP in fact has 2 Steiner vertices. Throughout this section, we extensively use the fact that cycles of SMNs cannot have chord-paths with less than 3 edges. Furthermore, we also use a straightforward fact that an SMN must have an even number of vertices of degree 3.
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Lemma 2. Let N be an SMN. If N has two adjacent vertices of degree 3, then N has at least 8 terminals. Proof. Assume that N has two adjacent vertices u and v of degree 3. Let a and b denote the other two vertices adjacent to u and let c and d denote the other two vertices adjacent to v. Let Cu denote a cycle through the edges (a, u) and (u, b). Let Cv denote a cycle through the edges (c, v) and (v, d). Cu and Cv must be distinct. If not, (u, v) would be a chord in Cu = Cv , contradicting the (length and degree) minimality of N . We claim that Cu and Cv must be disjoint. Assume that this is not the case. Let u denote the first vertex of Cu encountered when traversing Cv in one direction and let u denote the first vertex of Cu when traversing Cu in the other direction. u and u must be distinct; otherwise N would contain a vertex of degree 4, contradicting the minimality of N. The traversed paths from v to respectively u and u together with the portion of Cu between u and u containing u forms a cycle with (u, v) as a chord. This contradicts the minimality of N . From Corollary 1 it follows that N has at least 8 terminals. Lemma 3. Let N be an SMN. If N has two vertices of degree 3 adjacent to a common terminal of degree 2 then N has at least 9 terminals. Proof. The proof is analogous to the proof of Lemma 2. It exploits the fact that N cannot have chord-paths with 2 edges. Lemma 4. Let N be an SMN. If N has all degree 3 vertices separated by paths with at least 2 terminals of degree 2, then N has at least 6 terminals. Proof. Let u and v be two vertices of degree 3 in N . We can always choose u and v such that there is a path P between u and v containing only terminals of degree 2. Let a and b be the vertices adjacent to u not on P . N must contain a cycle Cu through the edges (a, u) and (u, b). Cu avoids any interior terminal of P . By Corollary 1, Cu contains 4 terminals. By assumption, P contains 2 terminals. Lemma 5. There exist problem instances where SMN has 6 terminals and 2 Steiner vertices.
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Fig. 4. SMN with 2 Steiner vertices and 6 terminals.
Proof. Consider the problem instance shown in Fig. 4. It is a complete graph K8 with 6 terminals (black circles) which have to be spanned and two additional vertices (white circles). All edges shown have unit lengths. All other edges have lengths equal to the lengths of shortest paths through unit edges. The set of edges shown forms a 2-connected subgraph of K8 spanning all 6 terminals. Hence, it is a feasible 2-connected solution. Its length is 9. We will show that this is the only optimal solution. Suppose that the solution shown in Fig. 4 is not optimal. Assume that an optimal solution contains no vertices of degree 3. Hence it is a traveling salesman tour with 6 edges. It has to contain all three unit length edges (A, D), (B, E) and (C, F ) connecting the terminals. Otherwise it would contain 4 or more edges of length at least 2 and would not be optimal. Hence, an optimal solution without degree 3 vertices is forced to contain 3 edges of length 2 connecting pairs of terminals. There are 6 such edges: (A, B), (A, C), (B, C), (D, E), (D, F ), (E, F ). It can be easily verified by inspection that no three of these edges will, together with (A, D), (B, E) and (C, F ), form a 2-connected solutions. The number of vertices of degree 3 in an SMN must be even. If SMN contains one Steiner vertex (and at least one terminal of degree 3), the total number of edges must be at least 8. At least 2 of the edges must have length at least 2. If SMN contains no Steiner vertices (and at least 2 terminals of degree 2), the total number of edges must be at least 7. At least 4 of the edges must have a length of at least 2.
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Steiner vertices. Replace each FST T in N by a minimum spanning tree (MST) for the set of terminals denote the resulting netspanned by T , and let N work. Note that N has no Steiner vertices, that is, all edge endpoints are terminals. Consider an edge (u, v) . Let PN (u, v) denote the unique path in N that in N ; if PN (u, v) consists of was replaced by (u, v) in N a single edge, it is identical to (u, v). We say that the edge (u, v) originates from the path PN (u, v). be defined as above. Then Lemma 6. Let N
Fig. 5. 2-connected SMNP in the Euclidean plane with 8 terminals and 2 Steiner vertices. This SMNP has two FSTs spanning three terminals and five FSTs spanning two terminals.
We conjecture that in the Euclidean plane with the standard L2 metric, no problem instance with 7 or less terminals has an SMNP with degree 3 vertices. In particular, such problem instances have no Steiner vertices and have traveling salesman tours as solutions to 2-SNPP. On the other hand, we have been able to construct a problem instance with 8 terminals which requires two Steiner vertices. Such an instance is shown in Fig. 5.
6. An algorithmic framework In this section, we present a novel algorithmic framework for solving 2-SNPG. The strength of this framework comes from the basic structural property stated by Theorem 2, namely all cycles contain at least four terminals. The main idea of this approach is to provide an upper bound on the number of Steiner vertices in any full Steiner tree (FST) of an SMN. More precisely, the algorithm determines a lower bound on the length of an SMN in which at least one FST has k or more Steiner vertices. If this lower bound exceeds an upper bound for the problem (e.g., the length of a traveling salesman tour), then there is no need to consider FSTs having k or more Steiner vertices. Let N be an SMN interconnecting n terminals, n 4. Assume that N contains an FST with k or more
is 2-connected, (i) N has at least 4 terminals, and (ii) every cycle in N has at least n + k edges. (iii) N Proof. (i) Any replacement of a FST in an SMN with a tree interconnecting the same terminals results in a network that remains 2-connected. This follows from the observation that no two vertex disjoint paths that connect a pair of terminals can use edges from the same FST [5]. By replacing every FST in N by its MST, we that is 2-connected. therefore obtain a network N (ii) Assume that N has a cycle with only two ter has no Steiner vertices, this minals u and v. Since N cycle consists of two (overlapping) edges (u, v) and (v, u). Let PN (u, v) and PN (v, u), respectively, be the paths that (u, v) and (v, u) originate from N . Other than meeting at their endpoints, these two paths must be disjoint, and since they only have Steiner vertices as interior vertices (if any), there is a cycle in N consisting of only two terminals—a contradiction to Theorem 2 (Fig. 6a). has a cycle with three termiNow assume that N nals u, v and x. Let (u, v), (v, x) and (x, u) be the three edges of the cycle, and let PN (u, v), PN (v, x) and PN (x, u) denote the paths that these edges originate from in N . If none of these paths have common vertices in their interior, there is a cycle in N with only three terminals, a contradiction to Theorem 2 (Fig. 6b). If at least two of the paths share interior vertices (i.e., are part of the same FST), then there is a cycle in N with only two terminals, again a contradiction (Fig. 6c). (iii) Let T be an FST in N with k or more Steiner vertices. Consider the network N \T . Luebke [5, Theorem 2.7] proved that there must be at least k vertices in N \T of degree 3. (The proof of this result is
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are drawn with tick lines, while edges from N are drawn with thin lines. Fig. 6. (a, b and c) Proof illustration for Lemma 6. Edges from N
quite involved and requires several definitions, otherwise not relevant for our purposes.) Let t denote the number of terminals in N \T that have degree 3, and let s denote the number of Steiner vertices in N \T (recall that all Steiner vertices have degree 3). Thus we have t + s k. The total degree of all terminals in N is at least : Each Steiner vertex in N —of 2n+t. Now consider N which there are at least s + k—increases the degree of some terminal by at least 1. Therefore, the total degree is at least 2n + t + s + k 2n + of all terminals in N 2k = 2(n + k). We will focus on solving the Euclidean 2-connected Steiner network problem in the plane (2-SNPP). Our approach can easily be used for 2-SNPG also, but is probably less powerful for this more general problem. For a given problem instance of 2-SNPP, let d(N) ) denote denote the length of an SMN N , and let d(N All FSTs in N are in N replaced by the length of N. MSTs. As a consequence of the Steiner ratio theorem [1], we have √2 d(N ). d(N) 3 k be a lower bound on the length of any network Let LB that fulfills the conditions stated in Lemma 6. Using the inequality above, this would mean that k √2 d(N ). LB 3 Let UB be an upper bound on d(N). If k > √2 U B LB 3
then we may conclude that N cannot have any FST with k or more Steiner vertices. Based on these observations, our algorithmic framework is as follows: 1. Compute an upper bound UB for the problem instance, e.g., a traveling salesman tour through the terminals. 2. Set k = 1. k using standard integer programming 3. Compute LB techniques, that is, a lower bound (or minimumlength solution) for a network satisfying the conditions given by Lemma 6. We used an integer programming formulation and branch-and-cut algorithm similar to the one discussed in [2,8]. k √2 U B then set k = k + 1 and go to 3. 4. If LB 3 5. Let k ∗ = k − 1. Compute an SMN for which every FST has at most k ∗ Steiner vertices. Experiments with this approach showed that it works well for two reasons. Firstly, for almost all problem instances, a traveling salesman tour is in fact an optimal solution to 2-SNPP. Thus the upper bound UB is very strong, in practice. Furthermore, conditions (ii) and (iii) of Lemma 6 fairly quickly make k increase as k increases. LB 7. Computational results and conclusions We performed a number of preliminary experiments with our new algorithmic framework. For randomly generated terminals in the plane, the results show that k ∗ is typically in the range 2–3 for n = 15, in the range 3–4 for n = 20 and range 4–6 for n = 25. This allows
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us to solve problem instances for n = 15 to optimality in seconds, whereas the exact algorithm of Luebke [5] used several hours to solve such problem instances. Problem instances of size n = 20 can be solved in minutes, and most problem instance of size n = 25 can be solved in less than 1 h. No randomly generated problem instances of size n > 15 have previously been solved to optimality. Further details of our exact algorithm and comprehensive computational experiments will be reported in [4]. The existence of an algorithm which can solve nontrivial problem instances with up to 30 terminals may prove useful in discovering additional structural properties of optimal solutions to 2-SNPP. References [1] D.Z. Du, F.K. Hwang, A proof of Gilbert and Pollak’s conjecture on the Steiner ratio, Algorithmica 7 (2/3) (1992) 121–135.
[2] M. Grötschel, C.L. Monma, M. Stoer, Design of survivable networks, in: Handbooks in Operations Research and Management Science, vol. 7, Elsevier Science, Amsterdam, 1995. [3] D.F. Hsu, X.-D. Hu, On shortest two-edge connected Steiner networks with Euclidean distance, Networks 32 (1998) 133– 140. [4] K.L. Hvam, L.B. Reinhardt, P. Winter, M. Zachariasen, Twoconnected Steiner networks, in preparation. [5] E.L. Luebke, K-connected Steiner network problems, Ph.D. Thesis, University of North Carolina, USA, 2002. [6] E.L. Luebke, J.S. Provan, On the structure and complexity of the 2-connected Steiner network problem in the plane, Oper. Res. Lett. 26 (2000) 111–116. [7] C.L. Monma, B.S. Munson, W.R. Pulleyblank, Minimumweight two-connected spanning networks, Math. Program. 46 (1990) 153–171. [8] D.M. Warme, Spanning trees in hypergraphs with applications to Steiner trees, Ph.D. Thesis, Computer Science Department, The University of Virginia, 1998.
Operations Research Letters www.elsevier.com/locate/dsw
Two-connected Steiner networks: structural properties Pawel Winter, Martin Zachariasen∗ Department of Computer Science, University of Copenhagen, Universitetsparken 1, DK-2100 Copenhagen ], Denmark Received 9 January 2004; accepted 29 July 2004 Available online 6 October 2004
Abstract We give a number of structural results for the problem of constructing a minimum-length 2-connected network for a set of terminals in a graph, where edge-weights satisfy the triangle inequality. A new algorithmic framework, based on our structural results, is given. © 2004 Elsevier B.V. All rights reserved. Keywords: Survivable networks; 2-connected Steiner networks
1. Introduction The well-known Steiner tree problem asks for a shortest possible network spanning a set Z of terminals in the plane. The solution to the problem is a tree, referred to as a Steiner minimal tree (SMT). Apart from the terminals that must be spanned, the SMT may contain additional, so-called Steiner points, where exactly three edges meet at 120◦ angles. SMTs are unions of full Steiner trees spanning subsets of terminals all having degree 1. When the objective is to design low-cost survivable networks, the problem of constructing 2-connected Steiner minimal networks in the plane arises. In a 2-edge-connected (resp. 2-vertex connected) Steiner ∗ Corresponding author.
E-mail addresses: [email protected] (P. Winter), [email protected] (M. Zachariasen). 0167-6377/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2004.07.010
network there are at least two edge-disjoint (resp. vertex-disjoint) paths between every pair of terminals. In the following, we do not distinguish between the vertex and edge connected variants of the problem for reasons to be explained in Section 2. The 2-connected Steiner network problem in the plane has been studied by Luebke and Provan [6], who proved that it is NP-hard and gave a number of structural properties. Additional structural properties for the generalized graph version where distances fulfill the triangle inequality were given by Luebke [5]. In this paper, we significantly improve and generalize the results given in [5,6]. We show that all cycles must have at least four terminals, and that each cycle must contain at least two non-incident edges connecting terminals. This allows us to give a tight lower bound on the number of terminals needed in any instance that requires a Steiner vertex in a minimumlength solution.
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The paper is organized as follows. In Section 2 we formally define the problems that are considered; we also review some of the known results. In Section 3 we prove some properties of the so-called chord-paths, and in Section 4 we give our main results on the distribution of terminals in cycles of 2-connected Steiner networks. Lower bounds (and tight examples) on the size of Steiner networks that require Steiner vertices are given in Section 5. In Section 6 we present an algorithmic framework, based on the structural results given in this paper, for solving the problem. Preliminary computational results and concluding remarks are given in Section 7.
2. Preliminaries Let G = (V , E) be a complete undirected graph with a distance function defined on its vertices. For a pair of vertices u, v ∈ V , let d(u, v) denote the distance between u and v. The distance function is assumed to fulfill the requirements of a metric, i.e., it is non-negative, symmetric and satisfies the triangle inequality. Finally, let Z ⊆ V be a set of terminals; the remaining vertices V \Z are called Steiner vertices. The 2-connected Steiner network problem (2SNPG) is to find subgraph G = (V , E ) of G such that • G is 2-edge-connected, • Z ⊆ V , • G has the minimum total length. Since a 2-edge-connected minimum-length network necessarily is 2-vertex-connected when the distance function is a metric [7], we use the shorthand 2connected in the following. This problem was studied by Monma et al. [7], where the main focus was on the special case when the graph contains no Steiner vertices, that is, when Z = V . They proved that there always exists an optimal solution in which all terminals have degree 2 or 3, and all Steiner vertices have degree 3. Hsu and Hu [3] and Luebke and Provan [6] considered a special case of the graph problem, namely the Euclidean 2-connected Steiner network problem in the plane (2-SNPP). In this problem, the terminals Z are points in the plane, and the task is to construct
a minimum-length 2-connected network that interconnects Z. (The problem is a special case of the graph problem since only a finite set of Steiner vertices needs to be considered due to the 120◦ angle conditions at Steiner vertices.) For this problem it was proved that Steiner vertices are incident to three edges meeting at 120◦ angles (as for the Euclidean Steiner tree problem in the plane), and that no cycles in a minimum-length network consist entirely of Steiner vertices. As a consequence, a shortest network is a union of full Steiner trees (FSTs), in which all terminals are leaves and all Steiner vertices are interior vertices. In this paper, we study the structure of minimumlength solutions for the general graph problem (2SNPG)—which contains 2-SNPP as a special case. More specifically, we will study the structure of minimum-length solutions in which the total degree of all vertices is minimized. In this way we avoid trivial degeneracies. We denote by SMN, an arbitrary minimum-length 2-connected network for which the total degree of all vertices has been minimized. (For 2-SNPP an arbitrary 2-connected minimum-length network is denoted by SMNP). Luebke [5] proved that the following properties hold for any SMN: • All vertices have degree 2 or 3, and all Steiner vertices have degree 3. • All edges are of multiplicity one (unless |Z| = 2). • No cycle is composed entirely of Steiner vertices. • For |Z| 4 there are no cycles with exactly three vertices.
3. Chord-path properties Let G denote an arbitrary undirected graph. Given a cycle C and two distinct vertices u and v on C in G, a chord-path between u and v is a path P (u, v) in G between u and v that, except from u and v, shares neither vertices nor edges with C. Note that the interior vertices in P (u, v) are not required to have degree 2 in G. When P (u, v) consists of a single edge, the chordpath reduces to a simple chord edge of C. It is well known [5,7] that an SMN cannot have any chord edge: Consider a cycle C having a chord edge (u, v). Clearly, the subgraph C ∪(u, v) is 2-connected. If we delete the edge (u, v) from this subgraph, the total length of the network does not increase and the
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total degree decreases. Furthermore, since the remaining subgraph (which is C) is still 2-connected, the resulting overall network remains 2-connected [7]. As a consequence, any chord-path in an SMN must have at least two edges. The following strengthens this result. Lemma 1. Any chord-path in an SMN must have at least three edges. Proof. Assume that there exists a cycle C with a chord-path P (u, v) consisting of two edges (u, t) and (t, v). Let a and b be the neighbours of u on C, and let c and d be the neighbours of v on C (see Fig. 1(a)). Neither vertices a and d nor vertices b and c are required to be distinct. Assume w.l.o.g. that d(t, v) d(t, u). Remove the edges (t, u) and (v, c), and add the edge (t, c), as shown in Fig. 1(b). The network remains 2-connected and its length does not increase: d(t, c) d(t, v) + d(v, c) d(t, u) + d(v, c). Both vertices u and v have reduced their degree with one, while the degree of all other vertices is unchanged. Thus we have arrived at a contradiction to the (length and degree) minimality of SMN. Luebke and Provan [6] showed for 2-SNPP that an SMNP cannot have a chord-path that has only Steiner vertices in its interior. Using Lemma 1, we can now generalize this result. Theorem 1. Any chord-path in an SMN must have a pair of consecutive terminals of degree 2 in its interior. Proof. Consider a chord-path with no consecutive terminals of degree 2 in its interior; among all such chord-
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paths, let P (u, v) be one with the minimum number of edges, and let C be a cycle for which P (u, v) is a chord-path. By Lemma 1, P (u, v) has at least two interior vertices, and hence at least one interior vertex w of degree 3—otherwise P (u, v) would have two consecutive terminals of degree 2 in its interior. Let x be the third vertex adjacent to w and not on P (u, v) (Fig. 2a). Edge (w, x) must be on some cycle Cw in SMN [7]. The cycle Cw uses one of the edges incident to w on P (u, v), since w has degree 3. As a consequence, Cw will share at least one vertex with C ∪ P (u, v), and distinct from w. Follow the cycle Cw , starting in x and moving away from w. At some point the cycle Cw will encounter a vertex y = w on C ∪ P (u, v); note that vertex y is distinct from w as argued above. Assume first that y is some vertex on P (u, v). Then the subpath of P (u, v) from w to y is a chord-path of another cycle in SMN, as shown in Fig. 2b. This chordpath has fewer edges than P (u, v). Furthermore, it has no pair of consecutive degree 2 terminals in its interior. This is a contradiction to the choice of P (u, v). Now assume that y is a vertex on C, distinct from u and v. The subpath of P (u, v) from u to w is a chordpath of another cycle, as shown in Fig. 2c. Again, this chord-path has fewer edges than P (u, v) and has no pair of consecutive degree 2 terminals in its interior. Once again this is a contradiction to the choice of P (u, v). In conclusion, chord-path P (u, v) cannot exist.
4. Cycle properties In this section, we use the properties of chord-paths to prove a number of fundamental structural properties for cycles in solutions to 2-SNPG. Theorem 2. Consider a cycle C that contains a vertex of degree 3 in an SMN. Then C must have two pairs of consecutive terminals, both of degree 2 in SMN. Furthermore, these two terminal pairs must be separated on C by a pair vertices of degree 3. Proof. Let w be a vertex on C of degree 3. Let x be the vertex adjacent to w not on C. Edge (w, x) must be on some cycle Cw in SMN; this cycle uses one of
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the edges on C incident to w (since w has degree 3). As a consequence, Cw will share at least one vertex with C, and is distinct from w. Follow the cycle Cw , starting in x and moving away from w. Let y be the first vertex on C that is encountered, and define P1 to be the path from w to y following the cycle Cw . Let P2 and P3 denote the two edge-disjoint paths from y to w on C. Note that the paths P1 , P2 and P3 share no vertices except from their endpoints. Consider the cycle P1 ∪ P2 . The path P3 is a chord-path of this cycle and by Theorem 1 it has a pair of consecutive terminals of degree 2 in SMN. Now consider the cycle P1 ∪ P3 . The path P2 is a chord-path of this cycle and must also have a pair of consecutive terminals of degree 2 in SMN. The theorem follows. Consider an instance of 2-SNPG with at least four terminals. Either the SMN for this problem is a simple cycle through all terminals, or every cycle has two vertices of degree 3. In both cases we get the following: Corollary 1. If the total number of terminals in an SMN is at least four, then every cycle in the SMN has at least four terminals of degree 2. An SMN is a union of full Steiner trees (FSTs) [6]. Consider a cycle C in an SMN in which every edge is from an FST spanning at least three terminals (Fig. 3). Thus every terminal in C has two Steiner vertices as neighbors, that is, cycle C has no pair of consecutive terminals of degree 2 as required by Theorem 2. We have the following corollary: Corollary 2. No cycle in an SMN has edges solely from full Steiner trees spanning three or more terminals.
Fig. 3. A cycle with edges solely from FSTs spanning three or more terminals. In this example, the cycle contains 6 terminals; the remaining vertices on the cycle are Steiner vertices in FSTs spanning three or more terminals.
5. Smallest networks with Steiner vertices In this section, we will show that an SMN cannot have vertices of degree 3 unless it contains at least 6 terminals. We will also show that this bound is tight for 2-SNPG. We conjecture that the smallest number of terminals needed for an SMNP (i.e., in the Euclidean plane) to have vertices of degree 3 is 8. We also give a problem instance with 8 terminals where the SMNP in fact has 2 Steiner vertices. Throughout this section, we extensively use the fact that cycles of SMNs cannot have chord-paths with less than 3 edges. Furthermore, we also use a straightforward fact that an SMN must have an even number of vertices of degree 3.
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Lemma 2. Let N be an SMN. If N has two adjacent vertices of degree 3, then N has at least 8 terminals. Proof. Assume that N has two adjacent vertices u and v of degree 3. Let a and b denote the other two vertices adjacent to u and let c and d denote the other two vertices adjacent to v. Let Cu denote a cycle through the edges (a, u) and (u, b). Let Cv denote a cycle through the edges (c, v) and (v, d). Cu and Cv must be distinct. If not, (u, v) would be a chord in Cu = Cv , contradicting the (length and degree) minimality of N . We claim that Cu and Cv must be disjoint. Assume that this is not the case. Let u denote the first vertex of Cu encountered when traversing Cv in one direction and let u denote the first vertex of Cu when traversing Cu in the other direction. u and u must be distinct; otherwise N would contain a vertex of degree 4, contradicting the minimality of N. The traversed paths from v to respectively u and u together with the portion of Cu between u and u containing u forms a cycle with (u, v) as a chord. This contradicts the minimality of N . From Corollary 1 it follows that N has at least 8 terminals. Lemma 3. Let N be an SMN. If N has two vertices of degree 3 adjacent to a common terminal of degree 2 then N has at least 9 terminals. Proof. The proof is analogous to the proof of Lemma 2. It exploits the fact that N cannot have chord-paths with 2 edges. Lemma 4. Let N be an SMN. If N has all degree 3 vertices separated by paths with at least 2 terminals of degree 2, then N has at least 6 terminals. Proof. Let u and v be two vertices of degree 3 in N . We can always choose u and v such that there is a path P between u and v containing only terminals of degree 2. Let a and b be the vertices adjacent to u not on P . N must contain a cycle Cu through the edges (a, u) and (u, b). Cu avoids any interior terminal of P . By Corollary 1, Cu contains 4 terminals. By assumption, P contains 2 terminals. Lemma 5. There exist problem instances where SMN has 6 terminals and 2 Steiner vertices.
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Proof. Consider the problem instance shown in Fig. 4. It is a complete graph K8 with 6 terminals (black circles) which have to be spanned and two additional vertices (white circles). All edges shown have unit lengths. All other edges have lengths equal to the lengths of shortest paths through unit edges. The set of edges shown forms a 2-connected subgraph of K8 spanning all 6 terminals. Hence, it is a feasible 2-connected solution. Its length is 9. We will show that this is the only optimal solution. Suppose that the solution shown in Fig. 4 is not optimal. Assume that an optimal solution contains no vertices of degree 3. Hence it is a traveling salesman tour with 6 edges. It has to contain all three unit length edges (A, D), (B, E) and (C, F ) connecting the terminals. Otherwise it would contain 4 or more edges of length at least 2 and would not be optimal. Hence, an optimal solution without degree 3 vertices is forced to contain 3 edges of length 2 connecting pairs of terminals. There are 6 such edges: (A, B), (A, C), (B, C), (D, E), (D, F ), (E, F ). It can be easily verified by inspection that no three of these edges will, together with (A, D), (B, E) and (C, F ), form a 2-connected solutions. The number of vertices of degree 3 in an SMN must be even. If SMN contains one Steiner vertex (and at least one terminal of degree 3), the total number of edges must be at least 8. At least 2 of the edges must have length at least 2. If SMN contains no Steiner vertices (and at least 2 terminals of degree 2), the total number of edges must be at least 7. At least 4 of the edges must have a length of at least 2.
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Steiner vertices. Replace each FST T in N by a minimum spanning tree (MST) for the set of terminals denote the resulting netspanned by T , and let N work. Note that N has no Steiner vertices, that is, all edge endpoints are terminals. Consider an edge (u, v) . Let PN (u, v) denote the unique path in N that in N ; if PN (u, v) consists of was replaced by (u, v) in N a single edge, it is identical to (u, v). We say that the edge (u, v) originates from the path PN (u, v). be defined as above. Then Lemma 6. Let N
Fig. 5. 2-connected SMNP in the Euclidean plane with 8 terminals and 2 Steiner vertices. This SMNP has two FSTs spanning three terminals and five FSTs spanning two terminals.
We conjecture that in the Euclidean plane with the standard L2 metric, no problem instance with 7 or less terminals has an SMNP with degree 3 vertices. In particular, such problem instances have no Steiner vertices and have traveling salesman tours as solutions to 2-SNPP. On the other hand, we have been able to construct a problem instance with 8 terminals which requires two Steiner vertices. Such an instance is shown in Fig. 5.
6. An algorithmic framework In this section, we present a novel algorithmic framework for solving 2-SNPG. The strength of this framework comes from the basic structural property stated by Theorem 2, namely all cycles contain at least four terminals. The main idea of this approach is to provide an upper bound on the number of Steiner vertices in any full Steiner tree (FST) of an SMN. More precisely, the algorithm determines a lower bound on the length of an SMN in which at least one FST has k or more Steiner vertices. If this lower bound exceeds an upper bound for the problem (e.g., the length of a traveling salesman tour), then there is no need to consider FSTs having k or more Steiner vertices. Let N be an SMN interconnecting n terminals, n 4. Assume that N contains an FST with k or more
is 2-connected, (i) N has at least 4 terminals, and (ii) every cycle in N has at least n + k edges. (iii) N Proof. (i) Any replacement of a FST in an SMN with a tree interconnecting the same terminals results in a network that remains 2-connected. This follows from the observation that no two vertex disjoint paths that connect a pair of terminals can use edges from the same FST [5]. By replacing every FST in N by its MST, we that is 2-connected. therefore obtain a network N (ii) Assume that N has a cycle with only two ter has no Steiner vertices, this minals u and v. Since N cycle consists of two (overlapping) edges (u, v) and (v, u). Let PN (u, v) and PN (v, u), respectively, be the paths that (u, v) and (v, u) originate from N . Other than meeting at their endpoints, these two paths must be disjoint, and since they only have Steiner vertices as interior vertices (if any), there is a cycle in N consisting of only two terminals—a contradiction to Theorem 2 (Fig. 6a). has a cycle with three termiNow assume that N nals u, v and x. Let (u, v), (v, x) and (x, u) be the three edges of the cycle, and let PN (u, v), PN (v, x) and PN (x, u) denote the paths that these edges originate from in N . If none of these paths have common vertices in their interior, there is a cycle in N with only three terminals, a contradiction to Theorem 2 (Fig. 6b). If at least two of the paths share interior vertices (i.e., are part of the same FST), then there is a cycle in N with only two terminals, again a contradiction (Fig. 6c). (iii) Let T be an FST in N with k or more Steiner vertices. Consider the network N \T . Luebke [5, Theorem 2.7] proved that there must be at least k vertices in N \T of degree 3. (The proof of this result is
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quite involved and requires several definitions, otherwise not relevant for our purposes.) Let t denote the number of terminals in N \T that have degree 3, and let s denote the number of Steiner vertices in N \T (recall that all Steiner vertices have degree 3). Thus we have t + s k. The total degree of all terminals in N is at least : Each Steiner vertex in N —of 2n+t. Now consider N which there are at least s + k—increases the degree of some terminal by at least 1. Therefore, the total degree is at least 2n + t + s + k 2n + of all terminals in N 2k = 2(n + k). We will focus on solving the Euclidean 2-connected Steiner network problem in the plane (2-SNPP). Our approach can easily be used for 2-SNPG also, but is probably less powerful for this more general problem. For a given problem instance of 2-SNPP, let d(N) ) denote denote the length of an SMN N , and let d(N All FSTs in N are in N replaced by the length of N. MSTs. As a consequence of the Steiner ratio theorem [1], we have √2 d(N ). d(N) 3 k be a lower bound on the length of any network Let LB that fulfills the conditions stated in Lemma 6. Using the inequality above, this would mean that k √2 d(N ). LB 3 Let UB be an upper bound on d(N). If k > √2 U B LB 3
then we may conclude that N cannot have any FST with k or more Steiner vertices. Based on these observations, our algorithmic framework is as follows: 1. Compute an upper bound UB for the problem instance, e.g., a traveling salesman tour through the terminals. 2. Set k = 1. k using standard integer programming 3. Compute LB techniques, that is, a lower bound (or minimumlength solution) for a network satisfying the conditions given by Lemma 6. We used an integer programming formulation and branch-and-cut algorithm similar to the one discussed in [2,8]. k √2 U B then set k = k + 1 and go to 3. 4. If LB 3 5. Let k ∗ = k − 1. Compute an SMN for which every FST has at most k ∗ Steiner vertices. Experiments with this approach showed that it works well for two reasons. Firstly, for almost all problem instances, a traveling salesman tour is in fact an optimal solution to 2-SNPP. Thus the upper bound UB is very strong, in practice. Furthermore, conditions (ii) and (iii) of Lemma 6 fairly quickly make k increase as k increases. LB 7. Computational results and conclusions We performed a number of preliminary experiments with our new algorithmic framework. For randomly generated terminals in the plane, the results show that k ∗ is typically in the range 2–3 for n = 15, in the range 3–4 for n = 20 and range 4–6 for n = 25. This allows
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us to solve problem instances for n = 15 to optimality in seconds, whereas the exact algorithm of Luebke [5] used several hours to solve such problem instances. Problem instances of size n = 20 can be solved in minutes, and most problem instance of size n = 25 can be solved in less than 1 h. No randomly generated problem instances of size n > 15 have previously been solved to optimality. Further details of our exact algorithm and comprehensive computational experiments will be reported in [4]. The existence of an algorithm which can solve nontrivial problem instances with up to 30 terminals may prove useful in discovering additional structural properties of optimal solutions to 2-SNPP. References [1] D.Z. Du, F.K. Hwang, A proof of Gilbert and Pollak’s conjecture on the Steiner ratio, Algorithmica 7 (2/3) (1992) 121–135.
[2] M. Grötschel, C.L. Monma, M. Stoer, Design of survivable networks, in: Handbooks in Operations Research and Management Science, vol. 7, Elsevier Science, Amsterdam, 1995. [3] D.F. Hsu, X.-D. Hu, On shortest two-edge connected Steiner networks with Euclidean distance, Networks 32 (1998) 133– 140. [4] K.L. Hvam, L.B. Reinhardt, P. Winter, M. Zachariasen, Twoconnected Steiner networks, in preparation. [5] E.L. Luebke, K-connected Steiner network problems, Ph.D. Thesis, University of North Carolina, USA, 2002. [6] E.L. Luebke, J.S. Provan, On the structure and complexity of the 2-connected Steiner network problem in the plane, Oper. Res. Lett. 26 (2000) 111–116. [7] C.L. Monma, B.S. Munson, W.R. Pulleyblank, Minimumweight two-connected spanning networks, Math. Program. 46 (1990) 153–171. [8] D.M. Warme, Spanning trees in hypergraphs with applications to Steiner trees, Ph.D. Thesis, Computer Science Department, The University of Virginia, 1998.