On some polynomially solvable cases and approximate algorithms in the optimal communication tree construction problem

Considering an arbitrary undirected n -vertex graph with nonnegative edge weights, we seek to construct a spanning tree minimizing the sum over all vertices of the maximal weights of the incident edges. We find some particular cases of polynomial solvability and show that the minimal span whose edge weights lie in the closed interval [ a, b ] is a -approximate solution, and the problem of constructing a 1.00048-approximate solution is NP-hard. We propose a heuristic polynomial algorithm and perform its a posteriori analysis.