The variation in which the diameter of the tree is bounded by a constant d the problem is not approximable within for any , unless NP , but approximable within for  and within for any constant . Admits a PTAS for the network Steiner tree problem if every element of S is adjacent to elements from V-S  and .
A prize-collecting variation in which a penalty is associated with each vertex and the goal is to minimize the cost of the tree and the vertices in S not in the tree is approximable within . The variation, called MINIMUM K-STEINER TREE, in which an integer is given in input and at least k vertices of S must be included in the subtree is approximable within 17 . Variation in which all non-required vertices are pairwise nonadjacent is approximable within 1.28 . Variation in which there are groups of required vertices and each group must be touched by the Steiner tree is approximable within , where g is the number of groups ; when the number of groups is unbounded the problem is harder than MINIMUM DOMINATING SET to approximate, even if all edge weights are 1 .
The constrained variation in which the input is extended with a positive integer k and a subset T of E, and the problem is to find the Steiner tree of weight at most k that contains the largest number of edges from T, is not approximable within for some . If the solution is allowed to be a forest with at most q trees, for a given constant q, the problem is approximable within . If the topology of the Steiner tree is given as input, the problem admits a PTAS . Finally, if before computing the Steiner tree the graph can be modified by reducing the weight of the edges within a given budget the problem of looking for the best reduction strategy is APX-hard .