Maximum shortest path interdiction problem by upgrading edges on trees under weighted $$l_1$$ norm

Abstract

Network interdiction problems by deleting critical edges have wide applicatio ns. However, in some practical applications, the goal of deleting edges is difficult to achieve. We consider the maximum shortest path interdiction problem by upgrading edges on trees (MSPIT) under unit/weighted $$l_1$$ norm. We aim to maximize the the length of the shortest path from the root to all the leaves by increasing the weights of some edges such that the upgrade cost under unit/weighted $$l_1$$ norm is upper-bounded by a given value. We construct their mathematical models and prove some properties. We propose a revised algorithm for the problem (MSPIT) under unit $$l_1$$ norm with time complexity O(n), where n is the number of vertices in the tree. We put forward a primal dual algorithm in $$O(n^2)$$ time to solve the problem (MSPIT) under weighted $$l_1$$ norm, in which a minimum cost cut is found in each iteration. We also solve the problem to minimize the cost to upgrade edges such that the length of the shortest path is lower bounded by a value and present an $$O(n^2)$$ algorithm. Finally, we perform some numerical experiments to compare the results obtained by these algorithms.