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

Shortest path with acceleration constraints: complexity and approximation algorithms

We introduce a variant of the Shortest Path Problem (SPP), in which we impose additional constraints on the acceleration over the arcs, and call it Bounded Acceleration SPP (BASP). This variant is inspired by an industrial application: a vehicle needs to travel from its current position to a target one in minimum-time, following pre-defined geometric paths connecting positions within a facility, while satisfying some speed and acceleration constraints depending on the vehicle position along the currently traveled path. We characterize the complexity of BASP, proving its NP-hardness. We also show that, under additional hypotheses on problem data, the problem admits a pseudo-polynomial time-complexity algorithm. Moreover, we present an approximation algorithm with polynomial time-complexity with respect to the data of the original problem and the inverse of the approximation factor ϵ. Finally, we present some computational experiments to evaluate the performance of the proposed approximation algorithm

Facility Location Planning Under Disruption

Facility Location Problems (FLPs) such as the Uncapacitated Facility Location (UFL) and the Capacitated Facility Location (CFL) along with the k-Shortest Path Problem (k-SPP) are important research problems in managing supply chain networks (SCNs) and related operations. In UFL, there is no limit on the facility serving capacity while in CFL such limit is imposed. FLPs aim to find the best facility locations to meet the customer demands within the available capacity with minimized facility establishment and transportation costs. The objective of the (k-SPP) is to find the k minimal length and partial overlapping paths between two nodes in a transport network graph. In the literature, many approaches are proposed to solve these problems. However, most of these approaches assume totally reliable facilities and do not consider the failure probability of the facilities, which can lead to notably higher cost. In this thesis, we investigate the reliable uncapacitated facility location (RUFL)and the reliable capacitated facility location (RCFL) problems, and the k-SPP where potential facilities are exposed to disruption then propose corresponding solution approaches to efficiently handle these problems. An evolutionary learning technique is elaborated to solve RUFL. Then, a non-linear integer programming model is introduced for the RCFL along with a solution approach involving the linearization of the model and its use as part of an iterative procedure leveraging CPLEX for facility establishment and customer assignment along with a knapsack implementation aiming at deriving the best facility fortification. In RUFL and RCFL, we assume heterogeneous disruption with respect to the facilities, each customer is assigned to primary and backup facilities and a fixed fortification budget allows to make a subset of the facilities totally reliable. Finally, we propose a hybrid approach based on graph partitioning and modified Dijkstra algorithm to find k partial overlapping shortest paths between two nodes on a transport network that is exposed to heterogeneous connected node failures. The approaches are illustrated via individual case studies along with corresponding key insights. The performance of each approach is assessed using benchmark results. For the k-SPP, the effect of preferred establishment locations is analyzed with respect to disruption scenarios, failure probability, computation time, transport costs, network size and partitioning parameters

Example-Based Urban Modeling

The manual modeling of virtual cities or suburban regions is an extremely time-consuming task, which expects expert knowledge of different fields. Existing modeling tool-sets have a steep learning curve and may need special education skills to work with them productively. Existing automatic methods rely on rule sets and grammars to generate urban structures; however, their expressiveness is limited by the rule-sets. Expert skills are necessary to typeset rule sets successfully and, in many cases, new rule-sets need to be defined for every new building style or street network style. To enable non-expert users, the possibility to construct urban structures for individual experiments, this work proposes a portfolio of novel example-based synthesis algorithms and applications for the controlled generation of virtual urban environments. The notion example-based denotes here that new virtual urban environments are created by computer programs that re-use existing digitized real-world data serving as templates. The data, i.e., street networks, topography, layouts of building footprints, or even 3D building models, necessary to realize the envisioned task is already publicly available via online services. To enable the reuse of existing urban datasets, novel algorithms need to be developed by encapsulating expert knowledge and thus allow the controlled generation of virtual urban structures from sparse user input. The focus of this work is the automatic generation of three fundamental structures that are common in urban environments: road networks, city block, and individual buildings. In order to achieve this goal, the thesis proposes a portfolio of algorithms that are briefly summarized next. In a theoretical chapter, we propose a general optimization technique that allows formulating example-based synthesis as a general resource-constrained k-shortest path (RCKSP) problem. From an abstract problem specification and a database of exemplars carrying resource attributes, we construct an intermediate graph and employ a path-search optimization technique. This allows determining either the best or the k-best solutions. The resulting algorithm has a reduced complexity for the single constraint case when compared to other graph search-based techniques. For the generation of road networks, two different techniques are proposed. The first algorithm synthesizes a novel road network from user input, i.e., a desired arterial street skeleton, topography map, and a collection of hierarchical fragments extracted from real-world road networks. The algorithm recursively constructs a novel road network reusing these fragments. Candidate fragments are inserted into the current state of the road network, while shape differences will be compensated by warping. The second algorithm synthesizes road networks using generative adversarial networks (GANs), a recently introduced deep learning technique. A pre- and postprocessing pipeline allows using GANs for the generation of road networks. An in-depth evaluation shows that GANs faithfully learn the road structure present in the example network and that graph measures such as area, aspect ratio, and compactness, are maintained within the virtual road networks. To fill empty city blocks in road networks we propose two novel techniques. The first algorithm re-uses real-world city blocks and synthesizes building footprint layouts into empty city blocks by retrieving viable candidate blocks from a database. We evaluate the algorithm and synthesize a multitude of city block layouts reusing real-world building footprint arrangements from European and US-cities. In addition, we increase the realism of the synthesized layouts by performing example-based placement of 3D building models. This technique is evaluated by placing buildings onto challenging footprint layouts using different example building databases. The second algorithm computes a city block layout, resembling the style of a real-world city block. The original footprint layout is deformed to construct a textit{guidance map}, i.e., the original layout is transferred to a target city block using warping. This guidance map and the original footprints are used by an optimization technique that computes a novel footprint layout along the city block edges. We perform a detailed evaluation and show that using the guidance map allows transferring of the original layout, locally as well as globally, even when the source and target shapes drastically differ. To synthesize individual buildings, we use the general optimization technique described first and formulate the building generation process as a resource-constrained optimization problem. From an input database of annotated building parts, an abstract description of the building shape, and the specification of resource constraints such as length, area, or a number of architectural elements, a novel building is synthesized. We evaluate the technique by synthesizing a multitude of challenging buildings fulfilling several global and local resource constraints. Finally, we show how this technique can even be used to synthesize buildings having the shape of city blocks and might also be used to fill empty city blocks in virtual street networks. All algorithms presented in this work were developed to work with a small amount of user input. In most cases, simple sketches and the definition of constraints are enough to produce plausible results. Manual work is necessary to set up the building part databases and to download example data from mapping services available on the Internet

Discrete particle swarm optimization for combinatorial problems with innovative applications.

Master of Science in Computer Science. University of KwaZulu-Natal, Durban 2016.Abstract available in PDF file