Efficient Benders decomposition for distance-based critical node detection problem

This paper addresses the critical node detection problem which seeks a subset of nodes for removal in order to maximize the disconnectivity of the residual graph with respect to a specific distance-based measure, namely the Wiener index. Such a measure is defined based on the all-pair shortest path distances in the residual graph so that the longer the total length of shortest paths, the greater the value of the disconnectivity measure. In the literature, a mixed integer linear programming model and an exact iterative-based method have been presented for this problem; however, both approaches become very time-consuming on graphs having large diameter and non-unit edge lengths. To overcome this shortcoming, in this paper, we present a new formulation for the problem and solve it by Benders decomposition algorithm. We improve the performance of Benders algorithm by several techniques (including analytical calculation of dual variables, generation of good-quality initial optimality cuts, considering master's optimality cuts as lazy constraints, etc.) to reduce the total running time. The extensive computational experiments on instances, taken from the literature or generated randomly, confirm the effectiveness of the new approaches.