The Steiner tree problem with hop constraints

The Steiner tree problem in graphs is to determine a minimum cost subgraph of a givengraph spanning a set of specified vertices. In certain telecommunication networks, additionalconstraints such as, e.g., reliability constraints, have to be observed. Assume that a certainreliability is associated with each arc of the network, measuring the probability that therespective arc is operational. In case there has to be a guarantee that each message sent froma root vertex to a specified vertex reaches its destination with a certain probability, so‐calledhop constraints may be used to model the respective generalization. In this paper, we discussthe Steiner tree problem with hop constraints, i.e., a generalization of Steiner's problem ingraphs where the number of arcs (hops) between a root node and any of the specified verticesis limited. A mathematical programming formulation is provided and extended to handleproblem instances of moderate size. As the Steiner tree problem with hop constraints is NP‐hard,a simple heuristic is developed and an exchange procedure based on the tabu searchmetastrategy is applied to improve given solutions. Numerical results are discussed for anumber of problem instances derived from, e.g., well‐known benchmark instances of Steiner'sproblem in graphs.