In this chapter, the inverse max+sum spanning tree problem, a novel class of inverse optimization problems with combined minimax-minsum objectives, has been thoroughly examined. By focusing on the modification of the sum-cost vector under both weighted \(l_1\) and \(l_\infty \) norms, as well as under weighted Hamming distance, we have developed mathematical models and efficient algorithms. The \(l_1\) norm problem has been transformed into a linear programming problem, addressed with a column generation algorithm, revealing a fascinating connection to the max+sum spanning tree problem, solvable in \(O(m \log n)\) time. The \(l_\infty \) norm problem has been proven to be a linear fractional combinatorial optimization problem, tackled with a discrete Newton method within \(O(m^2 \log m)\) iterations. For the Hamming distance, a binary search algorithm has been designed for the bottleneck case with a complexity of \(O(m \log ^2 n)\) , while the sum case has been identified as NP-hard. Modifying the max weight vector under the \(l_\infty \) norm led to a strongly polynomial time algorithm with a runtime of \(O(m^2 \log n)\) .