In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization. While Steiner tree problems may be formulated in a number of settings, they all require an optimal interconnect for a given set of objects and a predefined objective function. One well-known variant, which is often used synonymously with the term Steiner tree problem, is the Steiner tree problem in graphs. Given an undirected graph with non-negative edge weights and a subset of vertices, usually referred to as terminals, the Steiner tree problem in graphs requires a tree of minimum weight that contains all terminals (but may include additional vertices) and minimizes the total weight of its edges. Further well-known variants are the Euclidean Steiner tree problem and the rectilinear minimum Steiner tree problem. The Steiner tree problem in graphs can be seen as a generalization of two other famous combinatorial optimization problems: the (non-negative) shortest path problem and the minimum spanning tree problem. If a Steiner tree problem in graphs contains exactly two terminals, it reduces to finding the shortest path. If, on the other hand, all vertices are terminals, the Steiner tree problem in graphs is equivalent to the minimum spanning tree. However, while both the non-negative shortest path and the minimum spanning tree problem are solvable in polynomial time, no such solution is known for the Steiner tree problem. Its decision variant, asking whether a given input has a tree of weight less than some given threshold, is NP-complete, which implies that the optimization variant, asking for the minimum-weight tree in a given graph, is NP-hard. In fact, the decision variant was among Karp's original 21 NP-complete problems. The Steiner tree problem in graphs has applications in circuit layout or network design. However, practical applications usually require variations, giving rise to a multitude of Steiner tree problem variants. Most versions of the Steiner tree problem are NP-hard, but some restricted cases can be solved in polynomial time. Despite the pessimistic worst-case complexity, several Steiner tree problem variants, including the Steiner tree problem in graphs and the rectilinear Steiner tree problem, can be solved efficiently in practice, even for large-scale real-world problems.
Euclidean Steiner tree
The original problem was stated in the form that has become known as the Euclidean Steiner tree problem or geometric Steiner tree problem: Given N points in the plane, the goal is to connect them by lines of minimum total length in such a way that any two points may be interconnected by line segments either directly or via other points and line segments. While the problem is named after Steiner, it has first been posed in 1811 by Joseph Diez Gergonne in the following form: "A number of cities are located at known locations on a plane; the problem is to link them together by a system of canals whose total length is as small as possible". It may be shown that the connecting line segments do not intersect each other except at the endpoints and form a tree, hence the name of the problem. The problem for N = 3 has long been considered, and quickly extended to the problem of finding a star network with a single hub connecting to all of the N given points, of minimum total length. However, although the full Steiner tree problem was formulated in a letter by Gauss, its first serious treatment was in a 1934 paper written in Czech by Vojtěch Jarník and Miloš Kössler. This paper was long overlooked, but it already contains "virtually all general properties of Steiner trees" later attributed to other researchers, including the generalization of the problem from the plane to higher dimensions. For the Euclidean Steiner problem, points added to the graph (Steiner points) must have a degree of three, and the three edges incident to such a point must form three 120 degree angles (see Fermat point). It follows that the maximum number of Steiner points that a Steiner tree can have is N − 2, where N is the initial number of given points. (all these properties were established already by Gergonne.) For N = 3 there are two possible cases: if the triangle formed by the given points has all angles which are less than 120 degrees, the solution is given by a Steiner point located at the Fermat point; otherwise the solution is given by the two sides of the triangle which meet on the angle with 120 or more degrees. For general N, the Euclidean Steiner tree problem is NP-hard, and hence it is not known whether an optimal solution can be found by using a polynomial-time algorithm. However, there is a polynomial-time approximation scheme (PTAS) for Euclidean Steiner trees, i.e., a near-optimal solution can be found in polynomial time. It is not known whether the Euclidean Steiner tree problem is NP-complete, since membership to the complexity class NP is not known.
Rectilinear Steiner tree
The rectilinear Steiner tree problem is a variant of the geometric Steiner tree problem in the plane, in which the Euclidean distance is replaced with the rectilinear distance. The problem arises in the physical design of electronic design automation. In VLSI circuits, wire routing is carried out by wires that are often constrained by design rules to run only in vertical and horizontal directions, so the rectilinear Steiner tree problem can be used to model the routing of nets with more than two terminals.
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