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### MINIMUM STEINER TREE

• INSTANCE: Complete graph , a metric given by edge weights and a subset of required vertices.

• SOLUTION: A Steiner tree, i.e., a subtree of G that includes all the vertices in S.

• MEASURE: The sum of the weights of the edges in the subtree.

• Good News: Approximable within .

• Bad News: APX-complete .

• Comment: Variation in which the weights are only 1 or 2 is still APX-complete , but approximable within 1.28 . When all weights lie in an interval the problem is approximable within . On directed graphs the problem is approximable within for any .

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 .

• Garey and Johnson: ND12    Next: MINIMUM GEOMETRIC STEINER TREE Up: Spanning Trees Previous: MINIMUM COMMUNICATION COST SPANNING   Index
Viggo Kann
2000-03-20