In this paper, we propose a reinforcement pattern for the partial inverse minimum spanning tree problem, called the partial inverse minimum spanning tree problem with constant total weight constraint. Given a connected graph \(G=(V, E, w)\) and a forest F of G, the goal of this problem is to find a new weight function \(w^*\) , such that there exists a minimum spanning tree with respect to \(w^*\) containing F and the sum of weights of all edges remains unchanged. Meanwhile, we request the gap between w and \(w^*\) is minimum. In this paper, we study this problem under the bottleneck Hamming distance, and obtain its computational complexity. When \(\vert F\vert \geqslant 2\) , we show the inapproximability of it; when \(\vert F\vert =1\) , we present an algorithm with running time \(O(nm \log n)\) to solve it. In addition, for the special case where there is no restriction on the change of the weight function, we provide an algorithm with running time \(O(nm \log n)\) to solve it.
