In this chapter, we address the restricted inverse optimal value problem on minimum spanning tree under various norms. Under the weighted \(l_{\infty }\) norm with bounds, we establish optimality conditions and develop two efficient algorithms operating in \(O(m^2n)\) and \(O(m^2\log n)\) , where \(m\ \mbox{and}\ n\) are the number of edges and nodes of a graph G. Additionally, an \(O(mn)\) algorithm is introduced for the (RIOVMST \({ }_\infty \) ) problem with unit norm and bounds. For the \(l_1\) norm case, we formulate the problem as a linear program, derive a dual subproblem, and calculate a critical value \(z^*\) using binary search, ultimately solving the problem with a complexity of \(O(m^2n^2\log n\log (nC))\) . Under the bottleneck Hamming distance, three binary search algorithms are developed with a uniform time complexity of \(O(mn\log n)\) . An open problem is posed regarding a strongly polynomial time algorithm for the (RIOVMST \(_1\) ) problem.