We consider the problem of finding the most and the least "influential" or "influenceable" cliques in graphs based on three classical centrality measures: degree, closeness and betweenness. In addition to standard clique betweenness, which is defined as the proportion of shortest paths between any two graph nodes that pass through the clique, we also consider its optimistic and pessimistic versions, where outside nodes may favor or try to avoid shortest paths passing through the clique, respectively. We discuss the computational complexity issues for these problems and develop linear 0-1 programming formulations for each centrality metric. Finally, we demonstrate the performance of the developed formulations on real-life and synthetic networks, and provide some interesting insights based on the obtained results. In particular, our findings indicate that there are considerable variations between the centrality values of large cliques within the same networks. Moreover, the most central cliques in graphs are not necessarily the largest ones.
During natural or anthropogenic disasters, humanitarian organizations face a series of time-sensitive tasks.One of the tasks involves picking up critical resources (e.g., first aid kits, blankets, water) from warehouses and delivering them to the affected people. To successfully deliver these items to the people in need, the organization needs to make decisions that range from the quick acquisition of vehicles from the local market, to the preparation of pickup and delivery schedules and vehicle routes. During crises, the supply of vehicles is often limited, their acquisition cost is steep, and special rental periods are imposed. At the same time, the affected area needs the aid materials as fast as possible, and deliveries must be made within due time. Therefore, it is imperative that the decisions of acquiring, scheduling, and routing of vehicles are made optimally and quickly. In this paper, we consider a variant of a truckload open vehicle routing problem with time windows, which is suitable for modeling vehicle routing operations during a humanitarian crisis.We present two integer linear programming models to formulate the problem. The first one is an arc-based mixed integer linear programming model that is solved using a general purpose solver. The second one, on the other hand, is based on a path-based formulation, for which we design a column generation framework so as to solve it. Finally, we perform numerical experiments and compare the performance of the two models. The comparison shows that the latter path-based formulation outperforms the former without sacrificing solution quality when employing our column generation framework.
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