MaxSum Spanning Tree Interdiction and Improvement Problems Under Weighted l_∞ Norm⁺

The Max+Sum Spanning Tree (MSST) problem, with applications in secure communication systems, seeks a spanning tree T minimizing ${max}_{e\in T}w\left(e\right)+{\sum }_{e\in T}c\left(e\right)$ on a given edge-weighted undirected network $G(V,E,c,w)$, where the sets V and E are the sets of vertices and edges, respectively. The functions c and w are defined on the edge set, representing transmission cost and verification delay in secure communication systems, respectively. This problem can be solved within $O\left(\right|E|log|V\left|\right)$ time. We investigate its interdiction (MSSTID) and improvement (MSSTIP) problems under the weighted $l_{\infty }$ norm. MSSTID seeks minimal edge weight adjustments (to either c or w) to degrade network performance by ensuring the optimal MSST’s weight is at least K, while MSSTIP similarly aims to enhance performance by making the optimal MSST’s weight at most K through minimal weight modifications. These problems naturally arise in adversarial and proactive performance enhancement scenarios, respectively, where network robustness or efficiency must be guaranteed through constrained resource allocation. We first establish their mathematical models. Subsequently, we analyze the properties of the optimal value to determine the relationship between the magnitude of a given number and the optimal value. Then, utilizing binary search methods and greedy techniques, we design four algorithms with time complexity $O\left(\right|E{|}^{2}log\left|V\right|)$ to solve the above problems by modifying w or c. Finally, numerical experiments are conducted to demonstrate the effectiveness of the algorithms.