Mads Jepsen - Academia.edu (original) (raw)

Papers by Mads Jepsen

We initiate a systematic study of algorithms for optimization problems in the framework of differ... more We initiate a systematic study of algorithms for optimization problems in the framework of differential privacy, which formalizes the idea of protecting the privacy of individual input elements. We study the problems of vertex and set cover, min-cut, k-median, facility location, Steiner tree, and the recently introduced submodular maximization problem, Combinatorial Public Projects. For all these problems we give information theoretic lower bounds, and matching or nearly matching upper bounds.

This paper introduces a decomposition of the Elementary Shortest Path Problem with Resource Const... more This paper introduces a decomposition of the Elementary Shortest Path Problem with Resource Constraints(ESPPRC), where the path is combined by smaller sub paths. We show computationals result by comparing different approaches for the decomposition and compare the best of these with existing algorithms. We show that the algorithm for many instances outperforms a bidirectional labeling algorithm.

This paper presents a column generation algorithm for the Ve hicle Routing Problem with Time Wind... more This paper presents a column generation algorithm for the Ve hicle Routing Problem with Time Windows (VRPTW). Traditionally, column generation models of the VR PTW have consisted of a Set Partitioning master problem with each column representing a route, i.e., a resou ce feasible path starting and ending at the depot. Elementary routes (no customer visited more than once) have shown superior results on difficult instances (less restrictive capacity and time windows). However, the prici ng problems do not scale well when the number of feasible routes increases, i.e., when a route may contain a l arge number of customers. We suggest to relax that ‘each column is a route’ into ‘each column is a part of the gian t tour’; a so-called partial path, i.e., not necessarily starting and ending in the depot. This way, the length of the p artial path can be bounded and a better control of the size of the solution space for the pricing problem can be o btained.

Transportation Science, 2016

We consider an important generalization of the vehicle routing problem with time windows in which... more We consider an important generalization of the vehicle routing problem with time windows in which a fixed cost must be paid for accessing a set of edges. This fixed cost could reflect payment for toll roads, investment in new facilities, the need for certifications, and other costly investments. The certifications and investments impose a cost for the company while they also give unlimited usage of a set of roads to all vehicles belonging to the company. This violates the traditional assumption that the path between two destinations is well defined and independent of other choices. Different versions for defining the edge sets are discussed and formulated. Both the multigraph case and the direct path case are described, and mixed-integer-programming formulations of the problem are presented for both cases. A solution method based on branch-price-and-cut is applied to the direct path case. The computational results show that instances with up to 40 customers can be solved in a reason...

Discrete Optimization, 2014

ABSTRACT This paper considers the Capacitated Profitable Tour Problem (CPTP) which is a special c... more ABSTRACT This paper considers the Capacitated Profitable Tour Problem (CPTP) which is a special case of the Elementary Shortest Path Problem with Resource Constraints (ESPPRC). The CPTP belongs to the group of problems known as traveling salesman problems with profits. In CPTP each customer is associated with a profit and a demand and the objective is to find a capacitated tour (rooted in a depot node) that minimizes the total travel distance minus the profit of the visited customers. The CPTP can be recognized as the sub-problem in many column generation applications, where it is traditionally solved through dynamic programming. In this paper we present an alternative framework based on a formulation for the undirected CPTP and solved through branch-and-cut. Valid inequalities are presented among which we introduce a new family of inequalities for the CPTP denoted rounded multistar inequalities and we prove their validity. Computational experiments are performed on a set of instances known from the literature and a set of newly generated instances. The results indicate that the presented algorithm is highly competitive with the dynamic programming algorithms. In particular, we are able to solve instances with 800 nodes to optimality where the dynamic programming algorithms cannot solve instances with more than 200 nodes. Moreover dynamic programming and branch-and-cut complement each other well, giving us hope for solving more general problems through hybrid approaches. The paper is intended to serve as a platform for further development of branch-and-cut algorithms for CPTP hence also acting as a survey/tutorial.

Transportation Science, 2013

This paper presents an exact method for solving the symmetric two-echelon capacitated vehicle rou... more This paper presents an exact method for solving the symmetric two-echelon capacitated vehicle routing problem, a transportation problem concerned with the distribution of goods from a depot to a set of customers through a set of satellite locations. The presented method is based on an edge flow model that is a relaxation and provides a valid lower bound. A specialized branching scheme is employed to obtain feasible solutions. Out of a test set of 93 instances the algorithm is able to solve 47 to optimality, surpassing previous exact algorithms.

Operations Research, 2008

This paper presents a branch-and-cut-and-price algorithm for the vehicle-routing problem with tim... more This paper presents a branch-and-cut-and-price algorithm for the vehicle-routing problem with time windows. The standard Dantzig-Wolfe decomposition of the arc flow formulation leads to a set-partitioning problem as the master problem and an elementary shortest-path problem with resource constraints as the pricing problem. We introduce the subset-row inequalities, which are Chvatal-Gomory rank-1 cuts based on a subset of the constraints in the master problem. Applying a subset-row inequality in the master problem increases the complexity of the label-setting algorithm used to solve the pricing problem because an additional resource is added for each inequality. We propose a modified dominance criterion that makes it possible to dominate more labels by exploiting the step-like structure of the objective function of the pricing problem. Computational experiments have been performed on the Solomon benchmarks where we were able to close several instances. The results show that applying ...

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This note introduces an extension to the 0-1 knapsack cover inequalities to be used in a branch- ... more This note introduces an extension to the 0-1 knapsack cover inequalities to be used in a branch- and-cut algorithm for the elementary shortest path problem with a capacity constraint. The exten- sion leads to a set of valid inequalities that takes both the f ractional usage of the edges and the capacity into account and are denoted the flow extended 0-1 kn apsack cover inequalities. Compu- tational experiments indicate that although these new inequalities improve the lower bound they also results in more fractional LP solutions which results i n a larger number of branch nodes and eventually slower running times.

We consider an important generalization of the vehicle routing problem with time windows in which... more We consider an important generalization of the vehicle routing problem with time windows in which a fixed cost must be paid for accessing a set of edges. This fixed cost could reflect payment for toll roads, investment in new facilities, the need for certifications and other costly investments. The certifications and contributions impose a cost for the company while they also give unlimited usage of a set of roads to all vehicles belonging to the company. Different versions for defining the edge sets are discussed and formulated. A MIP-formulation of the problem is presented, and a solution method based on branch-and-price-and-cut is applied to the problem. The computational results show that instances with up to 50 customers can be solved in reasonable time, and that the branch-cut-and-price algorithm generally outperforms CPLEX. It also seems that instances get more difficult when the penalized edge sets form a spanning tree, compared to when they are randomly scattered.

Liner shipping networks are the backbone of international trade providing low transportation cost... more Liner shipping networks are the backbone of international trade providing low transportation cost, which is a major driver of globalization. These networks are under constant pressure to deliver capacity, cost effectiveness and environmentally conscious transport solutions. This article proposes a new path based MIP model for the Liner shipping Network Design Problem minimizing the cost of vessels and their fuel consumption facilitating a green network. The proposed model reduces problem size using a novel aggregation of demands. A decomposition method enabling delayed column generation is presented. The subproblems have similar structure to Vehicle Routing Problems, which can be solved using dynamic programming.

Transportation Science, 2016

Discrete Optimization, 2014

Transportation Science, 2013

Operations Research, 2008

We consider an important generalization of the vehicle routing problem with time windows in which... more We consider an important generalization of the vehicle routing problem with time windows in which a fixed cost must be paid for accessing a set of edges. This fixed cost could reflect payment for toll roads, investment in new facilities, the need for certifications and other costly investments. The certifications and contributions impose a cost for the company while they also give unlimited usage of a set of roads to all vehicles belonging to the company. Different versions for defining the edge sets are discussed and formulated. A MIP-formulation of the problem is presented, and a solution method based on branch-and-price-and-cut is applied to the problem. The computational results show that instances with up to 50 customers can be solved in reasonable time, and that the branch-cut-and-price algorithm generally outperforms CPLEX. It also seems that instances get more difficult when the penalized edge sets form a spanning tree, compared to when they are randomly scattered.