Three New Heuristics for the Steiner Problem in Graphs

Given a connected graph G = (V,E) (undirected, without loops and multiple edges) with positive edge costs (called also lengths) and a set Z ⊂ V of special (distinguished) vertices, the Steiner problem on graphs (networks) asks for a minimum cost tree within G that spans all members of Z. If |Z| = 2 we have the shortest path problem and if Z = V we get the minimum spanning tree problem, which are well known problems solvable in polynomial time. The same is true for any fixed cardinality p := |Z|. However, in general, the Steiner problem is NP-hard. Nevertheless, a tree that is not more than 2− 2/p times as expensive as an optimal tree can be computed in polynomial time. On the other hand no polynomial time approximation algorithm is known to have worst-case performance that is bounded by 2− ε times the cost of an optimal tree, for ε > 0. The Steiner problem has an extensive literature and numerous applications, such as the design of integrated circuits and telephone networks. For good surveys on the Steiner problem see Hwang and Richards [3] and Winter [13]. Many exact and approximation methods have been developed for this NP-hard problem. There are also several graph polynomial time heuristics for the Steiner problem [13,3] and it is the purpose of this paper to present three new such heuristics which practically compare favorably to several known ones (including the spanning tree heuristic [1,6,9], the path heuristic [12] and the average distance heuristic [10,11]) and have the same theoretical worst-case performance. First, in Section 1 we give a heuristic based on the spanning tree heuristic. In Section 2 we present a heuristic which chooses vertices for a tree according to their sum of all distances to the special vertices. Since a vertex with the minimum sum is often called a median vertex, our heuristic is said to be median. The third heuristic deletes vertices with largest sum of distances and is called an antimedian heuristic. It is presented in Section 3.