Computational complexity of the 2-connected Steiner network problem in the ℓp plane
Introduction
Geometric Steiner network problems involve designing networks that interconnect a given set of points in the plane (terminals) such that the total length of the network, by which we mean the sum of the lengths of all edges in the network, is as small as possible. The resulting network is a tree, referred to as a minimum Steiner tree. The best known variants of this problem including the Euclidean, rectilinear and more general uniform-orientation Steiner tree problems have all been shown to be NP-hard, and NP-complete under a suitable discretisation [2]. The reasons these problems are hard to solve include the fact that the minimum Steiner tree may include nodes other than the terminals (referred to as Steiner points) and that the topology (or underlying graph structure) of the tree is not known beforehand. The difficulty in solving the minimum Steiner tree problem contrasts with that of the minimum spanning tree problem, which also involves interconnecting a given set of terminals by a minimum length network, but without the use of Steiner points. The minimum spanning tree problem can be solved in time for any metric, where n is the number of given terminals, using for example the well-known Prim's algorithm.
The 2-edge-connected version of the Steiner network problem adds a robustness constraint that requires that there must be at least two edge-disjoint paths in the network between every pair of terminals. This is also equivalent to saying that the network contains no bridge, or in other words contains no edge which if deleted would cause the network to become disconnected. Since a 2-edge-connected minimum-length network is necessarily 2-vertex-connected when the distance function is a metric [14], we use the shorthand “2-connected” throughout the remainder of this paper. Such a constraint, sometimes referred to as a ‘survivability’ constraint, has obvious applications in the design of telecommunication networks, power networks, and other infrastructure networks [10]. Consequently, 2-connected and 3-connected versions of the geometric Steiner network problem have been studied by a number of authors [6], [7], [8], [18].
We will now define the Steiner network problem more formally. Let X be a set of terminals in the plane. A spanning network on X is a connected network that spans all terminals in X. A Steiner network on X is a connected network that spans all terminals in X and also spans a (potentially empty) finite set S of points anywhere in the plane. The points in S are called Steiner points. The geometric 2-connected Steiner network problem on X requires one to find a shortest length Steiner network on X, where length is measured according to a given norm. There may be additional constraints on the resultant network such as connectivity constraints. An optimal solution is referred to as a minimum Steiner network. As a decision problem, the problem we address can now be stated as follows. Let be a constant real number or let .
Geometric 2-connected Steiner network decision problem
Instance: A finite set of points X in the plane and a positive integer L.
Question: Is there a 2-connected Steiner network N with terminal set X such that the length of N (with respect to the metric) is at most L?
Our aim in this paper is to show that this problem is NP-hard, or indeed NP-complete if the problem is suitably discretised, when or . It remains unknown if the problem is NP-hard when . An attempt was made to prove the result for (i.e., in the Euclidean plane) by Luebke and Provan in [13]; however, their proof relies on a constant (the so-called Steiner ratio) which, in fact, currently remains unknown (a published proof for the value of this constant was shown to be incorrect; see [9]).
As a consequence of our method of proof, we will also show that the 2-connected spanning network problem is NP-hard when or . In other words, as opposed to the Steiner tree problem, the 2-connected Steiner network problem remains NP-hard (and NP-complete when suitably discretised) even when Steiner points are not permitted.
The organisation of this paper is as follows. In Section 2 we present some key background and preliminary results relating to both the global and local structure of 2-connected Steiner networks. In Section 3, we state our main complexity theorems specifying the NP-hardness of the two problems defined above, and detail the construction used to reduce any instance of the problem of deciding the Hamiltonicity of a planar 2-connected cubic bipartite graph to a suitable 2-connected Steiner network decision problem (recall that a cubic graph is a graph where every node is of degree three, and a bipartite graph is a graph with a node-set that can be partitioned into two independent sets). In Sections 4 and 5 we prove certain lower bounds which are crucial to our complexity proofs, and complete our main NP-hardness proof in Section 6. Finally, in Section 7 we extend the proof to a discretised version of the problem, thereby proving NP-completeness.
Section snippets
Full Steiner components
Let N be a minimum 2-connected Steiner network in a given normed plane. The edges of N can be uniquely partitioned into a set with the following properties:
- 1.
Each is non-empty
- 2.
Each is maximal with respect to the property that every pair of distinct edges in can be connected by a path in N each of whose interior vertices are Steiner points.
We refer to each as a full component of N. If is a tree then each terminal has degree 1 and we refer to as a full Steiner tree (for
The polynomial time reduction
As stated in the Introduction, the principal aim of this paper is to show that the geometric 2-connected Steiner network decision problem is NP-hard for all planes with or . We will not consider the case separately since it immediately follows from the case (observe that the 1-norm unit ball is simply a scaling and rotation of the ∞-norm unit ball). As a consequence of our proof, we will show that even without Steiner points (that is, when the constructed network spans
Lower bounds on distances between terminals in X
We next prove some key properties of the terminals in X which will be important for establishing bounds on the length of the minimum 2-connected Steiner network on X. In the following lemmas and their proofs note that the locations of all terminals (other than the centre points and tips) are dependent on the choice of p, as are the quantities z and q from Lemma 7.
Lemma 15 The shortest distance between two non-consecutive non-centre terminals in X is larger than , for some which depends on p
Lower bounds on the lengths of full components
We now prove a number of supplementary lemmas which gives bounds on the lengths of full components of a minimum 2-connected Steiner network on X. We use the notation to denote the length of any geometric graph A, by which we mean the sum of the lengths of all edges of A
Let N be a minimum 2-connected Steiner network on X and let T be any full component of N. Let be the number of terminals of T that are on a T-shape and let be the number of centre terminals contained in T.
Proof of Lemma 14
In this section we provide a proof of Lemma 14. Let N be an optimal 2-connected Steiner network on the set X of terminals constructed in Section 3. In proving the lemma we show that N consists of all possible small edges and exactly two centre-tip edges incident to each centre node if and only if G contains a Hamiltonian cycle. Now, let be a network that contains exactly this set of edges. Note that has length since there are n centre-tip edges of length 3n. Note also that
Discretisation
In this final section we show that the following discretised version of the 2-connected Steiner network problem in planes is NP-complete when or , thereby proving Theorem 11 and Corollary 13. In the discrete metric the length of any edge e is calculated as . We assume that the quantity can be calculated in constant time (by, for instance, assuming that p is rational).
Discrete Geometric 2-connected Steiner network decision problem
Instance: A finite set of points Y with
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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