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The problem of calculating the shortest path that visits a given set of nodes is at least as difficult as the traveling salesman problem, and it has not received much attention. Nevertheless an efficient integer linear programming (ILP) formulation has been recently proposed for this problem. That ILP formulation is firstly adapted to include the constraint that the obtained path can be protected by a node-disjoint path, and secondly to obtain a pair of node disjoint paths, of minimal total additive cost, each having to visit a given set of specified nodes. Computational experiments show that these approaches, namely in large networks, may fail to obtain solutions in a reasonable amount of time. Therefore heuristics are proposed for solving those problems, that may arise due to network management constraints. Extensive computational results show that they are capable of finding a solution in most cases, and that the …