An accelerating algorithm for maximum shortest path interdiction problem by upgrading edges on trees under unit Hamming distance

Abstract

Zhang et al. (Optim. Lett. 15(21):2661–2680, 2021) investigated the maximum shortest path interdiction problem by upgrading edges on trees under unit Hamming distance, and gave the algorithm with time complexity $O(N+l\log l)$ when $K=1$ and the algorithm with time complexity $O(N(\log N+K^3))$ when $K>1$, where N, K and l are the number of tree nodes, the number of upgraded edges and leaves respectively. In this paper, we optimize Zhang et al.’s results and give a greedy algorithm with time complexity O(N) and a dynamic programming algorithm with time complexity $O(NK^2)$. Meanwhile, based on this algorithm, we solve this problem with time complexity of $O(N^3\log N)$ through binary search by upgrading the edges on the tree under the unit Hamming distance mentioned by Zhang et al. for the minimum cost shortest path interdiction problem. In addition, each edge can only be updated at most once in Zhang et al.’s model. Then, we consider a maximum shortest path interdiction problem by multiple upgrading edges on trees under gradient Hamming distance (MSPITMGH). And we extend Zhang et al.’s study by proposing a new model, in which each edge can be upgraded multiple times and the weight of edges can be dynamically increased step by step, however, such model cannot be solved by Zhang et al.’s algorithm. For the MSPITMGH, an algorithm with time complexity of $O(NK^3)$ is given. Finally, we design numerical experiments to verify the correctness and efficiency of the proposed algorithms.