An adaptive discretization method for the shortest path problem with time windows

The Shortest Path Problem with Time Windows (SPPTW) is an important generalization of the classical shortest path problem. SPPTW has been extensively studied in practical problems, such as transportation optimization, scheduling, and routing problems. It also appears as a sub-problem in the column-generation process of the vehicle routing problem with time windows. In SPPTW, we consider a time-constrained graph, where each node is assigned with a time window, each edge is assigned with a cost and a travel time. The objective is to find the shortest path from a source node to a destination node while respecting the time window constraints. When the graph contains negative cycles, the problem becomes Elementary Shortest Path Problem with Time Windows (ESPPTW). In this thesis, we adopt the time-expanded network approach, extend it by incorporating the adaptive expansion idea and propose a new approach: Adaptive Time Window Discretization(ATWD) method. We demonstrate that the ATWD method can be easily combined with label setting algorithms and label correcting algorithms for solving SPPTW. We further extend the ATWD embedded label correcting algorithm by adding k-cycle elimination to solve ESPPTW on graphs with negative cycles. We also propose an ATWD based integer programming solution for solving ESPPTW. The objective of our study is to show that optimal solutions in a time-constrained network can be found without first constructing the entire time-expanded network

Variants of Shortest Path Problems

The shortest path problem in which the (s, t) -paths P of a given digraph G = (V, E) are compared with respect to the sum of their edge costs is one of the best known problems in combinatorial optimization. The paper is concerned with a number of variations of this problem having different objective functions like bottleneck, balanced, minimum deviation, algebraic sum, k -sum and k -max objectives, (k 1, k 2) -max, (k 1, k 2) -balanced and several types of trimmed-mean objectives. We give a survey on existing algorithms and propose a general model for those problems not yet treated in literature. The latter is based on the solution of resource constrained shortest path problems with equality constraints which can be solved in pseudo-polynomial time if the given graph is acyclic and the number of resources is fixed. In our setting, however, these problems can be solved in strongly polynomial time. Combining this with known results on k -sum and k -max optimization for general combinatorial problems, we obtain strongly polynomial algorithms for a variety of path problems on acyclic and general digraphs

Path planning algorithm for a car like robot based on MILP method

This project is presents an algorithm for path planning optimal routes mobile robot “like a car” to a target in unknown environment. The proposed algorithm allows a mobile robot to navigate through static obstacles and finding the path in order to reach the target without collision. This algorithm provides the robot the possibility to move from the initial position to the final position (target). The proposed path finding strategy is to use mathematical programming techniques to find the optimal path between to state for mobile robot designed in unknown environment with stationary obstacles. Formulation of the basic problems is to have the vehicle moved from the initial dynamic state to a state without colliding with each other, while at the same time avoiding other stationary obstacles. It is shown that this problem can be rewritten as a linear program with mixed integer / linear constraints that account for the collision avoidance. This approach is that the path optimization can be easily solved using the CPLEX optimization software with AMPL interface / MATLAB. The final phases are the design and build coalitions of linear programs and binary constraints to avoid collision with obstacles by Integer Mixed Linear Program (MILP). The findings of this research have shown that the MILP method can be used in the path planning problem in terms of finding a safe and shortest path. This has been combined with collision avoidance constraints to form a mixed integer linear program, which can be solved by a commercial software package

Problemas do caminho mais curto com restrições adicionais

Mestrado em Matemática e AplicaçõesNeste trabalho estudam-se problemas do caminho mais curto com restrições adicionais. Este tipo de problemas tem variadas aplicações práticas onde é destacado o planeamento de rotas de veículos e o encaminhamento de mensagens em redes de comunicações. O problema de caminho mais curto com restrições adicionais tem tido também grande aplicação como sub-problema de outros problemas. É o caso do problema de caminho mais curto com janelas temporais que surge como sub-problema do problema de determinação de rotas de veículos com janelas temporais. É feita uma descrição das várias variantes do problema de caminho mais curto com restrições adicionais, é apresentada uma revisão da literatura sobre os métodos usados na resolução deste tipo de problemas e são descritas algumas aplicações deste tipo de problemas.In this work, shortest path problems with additional constraints are studied. This type of problems has several practical applications, such as vehicle routing planning and the routing of messages in communications networks. The shortest path problem with additional constrains has also application as a sub-problem of other problems. This is the case of the shortest path problem with time windows occurring as sub-problem of the vehicle routing problem with time windows. We present a description of several variants of the shortest path problem with additional constraints, a short literature review on resolution methods for these problems and a description of some applications of this type of problems

Coverability in 1-VASS with Disequality Tests

We study a class of reachability problems in weighted graphs with constraints on the accumulated weight of paths. The problems we study can equivalently be formulated in the model of vector addition systems with states (VASS). We consider a version of the vertex-to-vertex reachability problem in which the accumulated weight of a path is required always to be non-negative. This is equivalent to the so-called control-state reachability problem (also called the coverability problem) for 1-dimensional VASS. We show that this problem lies in NC: the class of problems solvable in polylogarithmic parallel time. In our main result we generalise the problem to allow disequality constraints on edges (i.e., we allow edges to be disabled if the accumulated weight is equal to a specific value). We show that in this case the vertex-to-vertex reachability problem is solvable in polynomial time even though a shortest path may have exponential length. In the language of VASS this means that control-state reachability is in polynomial time for 1-dimensional VASS with disequality tests

Constrained shortest paths for QoS routing and path protection in communication networks.

The CSDP (k) problem requires the selection of a set of k > 1 link-disjoint paths with minimum total cost and with total delay bounded by a given upper bound. This problem arises in the context of provisioning paths in a network that could be used to provide resilience to link failures. Again we studied the LP relaxation of the ILP formulation of the problem from the primal perspective and proposed an approximation algorithm.We have studied certain combinatorial optimization problems that arise in the context of two important problems in computer communication networks: end-to-end Quality of Service (QoS) and fault tolerance. These problems can be modeled as constrained shortest path(s) selection problems on networks with each of their links associated with additive weights representing the cost, delay etc.The problems considered above assume that the network status is known and accurate. However, in real networks, this assumption is not realistic. So we considered the QoS route selection problem under inaccurate state information. Here the goal is to find a path with the highest probability that satisfies a given delay upper bound. We proposed a pseudo-polynomial time approximation algorithm, a fully polynomial time approximation scheme, and a strongly polynomial time heuristic for this problem.Finally we studied the constrained shortest path problem with multiple additive constraints. Using the LARAC algorithm as a building block and combining ideas from mathematical programming, we proposed a new approximation algorithm.First we studied the QoS single route selection problem, i.e., the constrained shortest path (CSP) problem. The goal of the CSP problem is to identify a minimum cost route which incurs a delay less than a specified bound. It can be formulated as an integer linear programming (ILP) problem which is computationally intractable. The LARAC algorithm reported in the literature is based on the dual of the linear programming relaxation of the ILP formulation and gives an approximate solution. We proposed two new approximation algorithms solving the dual problem. Next, we studied the CSP problem using the primal simplex method and exploiting certain structural properties of networks. This led to a novel approximation algorithm