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The Steiner tree problem, motorway problem, or minimum Steiner tree problem, named after Jakob Steiner, is a problem in combinatorial optimization, which may be formulated in a number of settings, with the common part being that it is required to find the shortest interconnect for a given set of objects. The Steiner tree problem is superficially similar to the minimum spanning tree problem: given a set ''V'' of points (vertices), interconnect them by a network (graph) of shortest length, where the length is the sum of the lengths of all edges. The difference between the Steiner tree problem and the minimum spanning tree problem is that, in the Steiner tree problem, extra intermediate vertices and edges may be added to the graph in order to reduce the length of the spanning tree. These new vertices introduced to decrease the total length of connection are known as Steiner points or Steiner vertices. It has been proved that the resulting connection is a tree, known as the Steiner tree. There may be several Steiner trees for a given set of initial vertices. The Steiner tree problem has applications in circuit layout or network design. Most versions of the Steiner tree problem are NP-complete. In fact, one of these was among Karp's original 21 NP-complete problems. Some restricted cases can be solved in polynomial time. In practice, heuristics are used. ==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. 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. 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. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Steiner tree problem」の詳細全文を読む スポンサード リンク
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