Positive and negative length-bound reachability constraints

In many application problems, including physical security and wildlife conservation, infrastructure must be configured to ensure or deny paths between specified locations. We model the problem as sub-graph design subject to constraints on paths and path lengths, and propose length-bound reachability constraints. Although reachability in graphs has been modelled before in constraint programming, the interaction of positive and negative reachability has not been studied in depth. We prove that deciding whether a set of positive and negative reachability constraints are satisfiable is NP complete. We show the effectiveness of our approach on decision problems, and also on optimisation problems. We compare our approach with existing constraint models, and we demonstrate significant improvements in runtime and solution costs, on a new problem set

Optimisation of large scale network problems

The Constrained Shortest Path Problem (CSPP) consists of finding the shortest path in a graph or network that satisfies one or more resource constraints. Without these constraints, the shortest path problem can be solved in polynomial time; with them, the CSPP is NP-hard and thus far no polynomial-time algorithms exist for solving it optimally. The problem arises in a number of practical situations. In the case of vehicle path planning, the vehicle may be an aircraft flying through a region with obstacles such as mountains or radar detectors, with an upper bound on the fuel consumption, the travel time or the risk of attack. The vehicle may be a submarine travelling through a region with sonar detectors, with a time or risk budget. These problems all involve a network which is a discrete model of the physical domain. Another example would be the routing of voice and data information in a communications network such as a mobile phone network, where the constraints may include maximum call delays or relay node capacities. This is a problem of current economic importance, and one for which time-sensitive solutions are not always available, especially if the networks are large. We consider the simplest form of the problem, large grid networks with a single side constraint, which have been studied in the literature. This thesis explores the application of Constraint Programming combined with Lagrange Relaxation to achieve optimal or near-optimal solutions of the CSPP. The following is a brief outline of the contribution of this thesis. Lagrange Relaxation may or may not achieve optimal or near-optimal results on its own. Often, large duality gaps are present. We make a simple modification to Dijkstra’s algorithm that does not involve any additional computational work in order to generate an estimate of path time at every node.We then use this information to constrain the network along a bisecting meridian. The combination of Lagrange Relaxation (LR) and a heuristic for filtering along the meridian provide an aggressive method for finding near-optimal solutions in a short time. Two network problems are studied in this work. The first is a Submarine Transit Path problem in which the transit field contains four sonar detectors at known locations, each with the same detection profile. The side constraint is the total transit time, with the submarine capable of 2 speeds. For the single-speed case, the initial LR duality gap may be as high as 30%. The first hybrid method uses a single centre meridian to constrain the network based on the unused time resource, and is able to produce solutions that are generally within 1% of optimal and always below 3%. Using the computation time for the initial Lagrange Relaxation as a baseline, the average computation time for the first hybrid method is about 30% to 50% higher, and the worst case CPU times are 2 to 4 times higher. The second problem is a random valued network from the literature. Edge costs, times, and lengths are uniform, randomly generated integers in a given range. Since the values given in the literature problems do not yield problems with a high duality gap, the values are varied and from a population of approximately 100,000 problems only the worst 200 from each set are chosen for study. These problems have an initial LR duality gap as high as 40%. A second hybrid method is developed, using values for the unused time resource and the lower bound values computed by Dijkstra’s algorithm as part of the LR method. The computed values are then used to position multiple constraining meridians in order to allow LR to find better solutions.This second hybrid method is able to produce solutions that are generally within 0.1% of optimal, with computation times that are on average 2 times the initial Lagrange Relaxation time, and in the worst case only about 5 times higher. The best method for solving the Constrained Shortest Path Problem reported in the literature thus far is the LRE-A method of Carlyle et al. (2007), which uses Lagrange Relaxation for preprocessing followed by a bounded search using aggregate constraints. We replace Lagrange Relaxation with the second hybrid method and show that optimal solutions are produced for both network problems with computation times that are between one and two orders of magnitude faster than LRE-A. In addition, these hybrid methods combined with the bounded search are up to 2 orders of magnitude faster than the commercial CPlex package using a straightforward MILP formulation of the problem. Finally, the second hybrid method is used as a preprocessing step on both network problems, prior to running CPlex. This preprocessing reduces the network size sufficiently to allow CPlex to solve all cases to optimality up to 3 orders of magnitude faster than without this preprocessing, and up to an order of magnitude faster than using Lagrange Relaxation for preprocessing. Chapter 1 provides a review of the thesis and some terminology used. Chapter 2 reviews previous approaches to the CSPP, in particular the two current best methods. Chapter 3 applies Lagrange Relaxation to the Submarine Transit Path problem with 2 speeds, to provide a baseline for comparison. The problem is reduced to a single speed, which demonstrates the large duality gap problem possible with Lagrange Relaxation, and the first hybrid method is introduced.Chapter 4 examines a grid network problem using randomly generated edge costs and weights, and introduces the second hybrid method. Chapter 5 then applies the second hybrid method to both network problems as a preprocessing step, using both CPlex and a bounded search method from the literature to solve to optimality. The conclusion of this thesis and directions for future work are discussed in Chapter 6

A hybrid column generation algorithm based on metaheuristic optimization

The exact solution and heuristic solution have their own strengths and weaknesses on solving the Vehicle Routing Problems with Time Windows (VRPTW). This paper proposes a hybrid Column Generation Algorithm with Metaheuristic Optimization (CGAMO) to overcome their weaknesses. Firstly, a Modified Labelling Algorithm (MLA) in the sub-problem of path searching is analysed. And a search strategy in CGAMO based on the demand of sub-problem is proposed to improve the searching efficiency. While putting the paths found in the sub-problem into the main problems of CGAMO, the iterations may fall into endless loops. To avoid this problem and keep the main problems in a reasonable size, two conditions on saving the old paths in the main problem are used. These conditions enlarge the number of constraints considered in the iterations to strengthen the limits of dual variables. Through analysing the sub-problem, we can find many useless paths that have no effect on the objective function. Secondly, in order to reduce the number of useless paths and improve the efficiency, this paper proposes a heuristic optimization strategy of CGAMO for dual variables. It is supposed to accelerate the solving speed from the view of on the dual problem. Finally, extensive experiments show that CGAMO achieves a better performance than other state-of-the-art methods on solving VRPTW. The comparative experiments also present the parameters sensitivity analysis, including the different effects of MLA in the different path selection strategies, the characteristics and the applicable scopes of the two pathkeeping conditions in the main problem. First published online: 25 Oct 201

Cost-Based Filtering for Shorter Path Constraints

Many real world problems, e.g. in personnel scheduling and transportation planning, can be modeled naturally as Constrained Shortest Path Problems (CSPPs), i.e., as Shortest Path Problems with additional constraints. A well studied problem in this class is the Resource Constrained Shortest Path Problem. Reduction technique

Proceedings of CSCLP 2007: Annual ERCIM Workshop on Constraint Solving and Constraint Logic Programming

Ce fichier regroupe en un seul document l'ensemble des articles acceptés pour la conférence CSCLP 2007Constraints are a natural way to represent knowledge, and constraint programming is a declarative programming paradigm that has been successfully used to express and solve many practical combinatorial optimization problems. Examples of application domains are scheduling, production planning, resource allocation, communication networks, robotics, and bioinformatics. These proceedings contain the research papers presented at the 12th International Workshop on Constraint Solving and Constraint Logic Programming (CSCLP'07), held on June 7th and 8th 2007, at INRIA Rocquencourt, France. This workshop, open to all, is organized as the twelfth meeting of the working group on Constraints of the European Research Consortium for Informatics and Mathematics (ERCIM). It continues a series of workshops organized since the creation of the working group in 1997, that have led since 2002 to the publication of a series of books entitled ”Recent Advances in Constraints” in the Lecture Notes in Artificial Intelligence, edited by Springer-Verlag. In addition to the contributed papers collected in this volume, two invited talks were given at CSCLP'07, one by Gilles Pesant, Ecole Polytechnique de Montreal, Canada, and one by Jean-Charles R égin, ILOG, France. The editors would like to take the opportunity to thank all the authors who submitted a paper, as well as the reviewers for their helpful work. CSCLP'07 has been made possible thanks to the support of the European Research Consortium for Informatics and Mathematics (ERCIM), the Institut National de la Recherche en Informatique et Automatique (INRIA) and the Association for Constraint programming (ACP)

Applications of matching theory in constraint programming

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