The binary identification problem for weighted trees

The Binary Identification Problem for weighted trees asks for the minimum cost strategy (decision tree) for identifying a vertex in an edge weighted tree via testing edges. Each edge has assigned a different cost, to be paid for testing it. Testing an edge e reveals in which component of T−e lies the vertex to be identified. We give a complete characterization of the computational complexity of this problem with respect to both tree diameter and degree. In particular, we show that it is strongly NP-hard to compute a minimum cost decision tree for weighted trees of diameter at least 6, and for trees having degree three or more. For trees of diameter five or less, we give a polynomial time algorithm. Moreover, for the degree 2 case, we significantly improve the straightforward O(n3) dynamic programming approach, and provide an O(n2) time algorithm. Finally, this work contains the first approximate decision tree construction algorithm that breaks the barrier of factor logn.