We give a survey up to 1989 on the Steiner tree problems which include the four important cases of euclidean, rectilinear, graphic, phylogenetic and some of their generalizations. We also provide a rather comprehensive and up-to-date bibliography which covers more than three hundred items.

Steiner tree problems

An iterative method is described which generates a minimal tree with a Steiner topology in at most n 2 steps, where n is the number of fixed vertices. The SI algorithm is formulated. When n < 4, the SI algorithm converges to a proper tree. Experimental studies indicate that this algorithm generates trees close to optimal Steiner minimal trees.

The Generation of Minimal Trees with a Steiner Topology

This note lists 226 papers published largely during the last ten years dealing with normative approaches to solving location problems.

Technical Note-Location Theory: A Selective Bibliography

Applications of Set Covering, Set Packing and Set Partitioning Models: A Survey

http://www.cs.concordia.ca/~chvatal/workshop06/steiner.pdf

Solving group Steiner problems as Steiner problems

1. Let Li , ■ ■ • , Ln be n separate lakes which are to be interconnected by a network of canals of minimum possible length. Or, let il/, , • • ■ , Mn be n metropolitan areas which are to be joined by the shortest possible network of roads. Again, let Py , • •• , P„ be n metal plates in the plane which are to be soldered together by the least possible amount of wire of fixed diameter. Assuming spatial homogeneity, we formulate an abstract problem which underlies the given examples. By a net N we shall understand a finite set of plane rectifiable arcs; the sum L(N) of their lengths is the length of N. Let n > 3 and let A = \Al , • • ■ , An\ be a plane set with exactly n components A{ ; it will be assumed throughout that each A { is compact. Any such A will be called an w-tcrminal set and its n components will be called terminals. Our problem is:

https://www.ams.org/qam/1968-26-02/S0033-569X-1968-0233286-9/S0033-569X-1968-0233286-9.pdf

E. J. Cockayne

Papers

Steiner’s problem for set-terminals