Inverse max sum spanning tree problem under Hamming distance by modifying the sum-cost vector

The inverse max sum spanning tree (MSST) problem is considered by modifying the sum-cost vector under the Hamming Distance. On an undirected network G(V, E, w, c), a weight w(e) and a cost c(e) are prescribed for each edge . The MSST problem is to find a spanning tree which makes the combined weight as small as possible. It can be solved in time, where and . Whereas, in an inverse MSST problem, a given spanning tree of G is not an optimal MSST. The sum-cost vector c is to be modified to so that becomes an optimal MSST of the new network and the cost can be minimized under Hamming Distance. First, we present a mathematical model for the inverse MSST problem and a method to check the feasibility. Then, under the weighted bottleneck-type Hamming distance, we design a binary search algorithm whose time complexity is . Next, under the unit sum-type Hamming distance, which is also called norm, we show that the inverse MSST problem (denoted by IMSST) is hard. Assuming , the problem IMSST is not approximable within a factor of , for any . Finally, We consider the augmented problem of IMSST (denoted by AIMSST), whose objective function is to multiply the norm by a sufficiently large number M plus the norm . We show that the augmented problem and the norm problem have the same Lagrange dual problems. Therefore, the norm problem is the best convex relaxation (in terms of Lagrangian duality) of the augmented problem AIMSST, which has the same optimal solution as that of the inverse problem IMSST.