Maximum coverage with cluster constraints: An LP-based approximation technique

Packing problems constitute an important class of optimization problems. However, despite the large number of variants that have been studied in the literature, most packing problems encompass a single tier of capacity restrictions only. For example, in the Multiple Knapsack Problem, we want to assign a selection of items to multiple knapsacks such that their capacities are not exceeded. But what if these knapsacks are partitioned into clusters, each imposing an additional (aggregated) capacity restriction on the knapsacks contained in that cluster? In this paper, we study the Maximum Coverage Problem with Cluster Constraints (MCPC), which generalizes the Maximum Coverage Problem with Knapsack Constraints (MCPK) by incorporating such cluster constraints. Our main contribution is a general LP-based technique to derive approximation algorithms for such cluster capacitated problems. Our technique basically allows us to reduce the cluster capacitated problem to the respective original packing problem. By using an LP-based approximation algorithm for the original problem, we can then obtain an effective rounding scheme for the problem, which only loses a small fraction in the approximation guarantee. We apply our technique to derive approximation algorithms for MCPC. To this aim, we develop an LP-based 12(1−1e) -approximation algorithm for MCPK by adapting the pipage rounding technique. Combined with our reduction technique, we obtain a 13(1−1e) -approximation algorithm for MCPC. We also derive improved results for a special case of MCPC, the Multiple Knapsack Problem with Cluster Constraints (MKPC). Based on a simple greedy algorithm, our approach yields a 13 -approximation algorithm. By combining our technique with a more sophisticated iterative rounding approach, we obtain a 12 -approximation algorithm for certain special cases of MKPC

Optimizing pre-processing and relocation moves in the Stochastic Container Relocation Problem

In container terminals, containers are often moved to other stacks in order to access containers that need to leave the terminal earlier. We propose a new optimization model in which the containers can be moved in two different phases: a pre-processing and a relocation phase. To solve this problem, we develop an optimal branch-and-bound algorithm. Furthermore, we develop a local search heuristic because the problem is NP-hard. Besides that, we give a rule-based method to estimate the number of relocation moves in a bay. The local search heuristic produces solutions that are close to the optimal solution. Finally, for instances in which the benefits of moving containers in the two different phases are in balance, the solution of the heuristic yields significant improvement compared to the existing methods in which containers are only moved in one of the two phases

Minimum costs paths in intermodal transportation networks with stochastic travel times and overbookings

In intermodal transportation, it is essential to balance the trade-off between the cost and duration of a route. The duration of a path is inherently stochastic because of delays and the possibility of overbooking. We study a problem faced by a company that supports shippers with advice for the route selection. The challenge is to find Pareto-optimal solutions regarding the route's costs and the probability of arriving before a specific deadline. We show how this probability can be calculated in a network with scheduled departure times and the possibility of overbookings. To solve this problem, we give an optimal algorithm, but as its running time becomes too long for larger networks, we also develop a heuristic. The idea of this heuristic is to replace the stochastic variables by deterministic risk measures and solve the resulting deterministic optimization problem. The heuristic produces, in a fraction of the optimal algorithm's running time, solutions of which the costs are only a few percent higher than the optimal costs

Léon le Grand. Claude Sarrasin, intendant, intendant des archives du chapitre de Notre-Dame de Paris, et sa collection d'extraits des registres capitulaires de Notre-Dame

Léon le Grand. Claude Sarrasin, intendant, intendant des archives du chapitre de Notre-Dame de Paris, et sa collection d'extraits des registres capitulaires de Notre-Dame. In: Bibliothèque de l'école des chartes. 1902, tome 63. p. 390