In this chapter, we delve into the inverse minimum spanning tree (IMST) problems under various norm constraints, including the \(l_1\) , \(l_\infty \) norms and the weighted Hamming distance. We present mathematical formulations and propose efficient algorithms tailored to each scenario. The bounded IMST problem under the unit \(l_1\) norm is reformulated as a maximum flow problem, which can be effectively solved within \(O(m^8 \log m)\) . In contrast, the unbounded IMST problem under the unit \(l_1\) norm is transformed into an unbalanced assignment problem, solvable in \(O(n^2m)\) time. Furthermore, this problem is also cast as a minimum cost flow problem, solvable in \(O(n^2 \log n)\) time. When considering the unbounded IMST problem under the weighted \(l_1\) norm, it is mapped onto an unbalanced transportation problem, with a solution time of \(O(n^2m\log (nc))\) , where \(C = \max \{|c_j| : a_j \in A\}\) . The unbounded IMST problem under the unit \(l_\infty \) norm is more straightforward, admitting a solution in \(O(n^2)\) time. For the bounded IMST problem under the weighted sum Hamming distance, we have transformed it into a minimum-weight node cover problem of a bipartite graph, which can be addressed in \(O(n^3m)\) time. Employing the binary method, the constrained bounded IMST problem under the weighted bottleneck Hamming distance is also transformed into a minimum-weight node cover problem of a bipartite graph, with a solution time of \(O(n^3m\log m)\) .

Inverse Minimum Spanning Tree Problems

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发表日期：

2025年

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Xiucui Guan,
Panos M. Pardalos,
Binwu Zhang

摘要

In this chapter, we delve into the inverse minimum spanning tree (IMST) problems under various norm constraints, including the \(l_1\) , \(l_\infty \) norms and the weighted Hamming distance. We present mathematical formulations and propose efficient algorithms tailored to each scenario. The bounded IMST problem under the unit \(l_1\) norm is reformulated as a maximum flow problem, which can be effectively solved within \(O(m^8 \log m)\) . In contrast, the unbounded IMST problem under the unit \(l_1\) norm is transformed into an unbalanced assignment problem, solvable in \(O(n^2m)\) time. Furthermore, this problem is also cast as a minimum cost flow problem, solvable in \(O(n^2 \log n)\) time. When considering the unbounded IMST problem under the weighted \(l_1\) norm, it is mapped onto an unbalanced transportation problem, with a solution time of \(O(n^2m\log (nc))\) , where \(C = \max \{|c_j| : a_j \in A\}\) . The unbounded IMST problem under the unit \(l_\infty \) norm is more straightforward, admitting a solution in \(O(n^2)\) time. For the bounded IMST problem under the weighted sum Hamming distance, we have transformed it into a minimum-weight node cover problem of a bipartite graph, which can be addressed in \(O(n^3m)\) time. Employing the binary method, the constrained bounded IMST problem under the weighted bottleneck Hamming distance is also transformed into a minimum-weight node cover problem of a bipartite graph, with a solution time of \(O(n^3m\log m)\) .