Network Models with Convex Cost Structure like Bundle Methods

For three rather diverse applications (truck scheduling for inter warehouse logistics, university-course timetabling, operational train timetabling) that contain integer multi-commodity flow as a major modeling element we present a computational comparison between a bundle and a full linear programming (LP) approach for solving the basic relaxations. In all three cases computing the optimal solutions with LP standard solvers is computationally very time consuming if not impractical due to high memory consumption while bundle methods produce solutions of sufficient but low accuracy in acceptable time. The rounding heuristics generate comparable results for the exact and the approximate solutions, so this entails no loss in quality of the final solution. Furthermore, bundle methods facilitate the use of nonlinear convex cost functions. In practice this not only improves the quality of the relaxation but even seems to speed up convergence of the method

Engineering Art Galleries

The Art Gallery Problem is one of the most well-known problems in Computational Geometry, with a rich history in the study of algorithms, complexity, and variants. Recently there has been a surge in experimental work on the problem. In this survey, we describe this work, show the chronology of developments, and compare current algorithms, including two unpublished versions, in an exhaustive experiment. Furthermore, we show what core algorithmic ingredients have led to recent successes

Exploiting Hopsets: Improved Distance Oracles for Graphs of Constant Highway Dimension and Beyond

For fixed h >= 2, we consider the task of adding to a graph G a set of weighted shortcut edges on the same vertex set, such that the length of a shortest h-hop path between any pair of vertices in the augmented graph is exactly the same as the original distance between these vertices in G. A set of shortcut edges with this property is called an exact h-hopset and may be applied in processing distance queries on graph G. In particular, a 2-hopset directly corresponds to a distributed distance oracle known as a hub labeling. In this work, we explore centralized distance oracles based on 3-hopsets and display their advantages in several practical scenarios. In particular, for graphs of constant highway dimension, and more generally for graphs of constant skeleton dimension, we show that 3-hopsets require exponentially fewer shortcuts per node than any previously described distance oracle, and also offer a speedup in query time when compared to simple oracles based on a direct application of 2-hopsets. Finally, we consider the problem of computing minimum-size h-hopset (for any h >= 2) for a given graph G, showing a polylogarithmic-factor approximation for the case of unique shortest path graphs. When h=3, for a given bound on the space used by the distance oracle, we provide a construction of hopset achieving polylog approximation both for space and query time compared to the optimal 3-hopset oracle given the space bound

Fast Robust Shortest Path Computations

We develop a fast method to compute an optimal robust shortest path in large networks like road networks, a fundamental problem in traffic and logistics under uncertainty. In the robust shortest path problem we are given an s-t-graph D(V,A) and for each arc a nominal length c(a) and a maximal increase d(a) of its length. We consider all scenarios in which for the increased lengths c(a) + bar{d}(a) we have bar{d}(a) <= d(a) and sum_{a in A} (bar{d}(a)/d(a)) <= Gamma. Each path is measured by the length in its worst-case scenario. A classic result [Bertsimas and Sim, 2003] minimizes this path length by solving (|A| + 1)-many shortest path problems. Easily, (|A| + 1) can be replaced by |Theta|, where Theta is the set of all different values d(a) and 0. Still, the approach remains impractical for large graphs. Using the monotonicity of a part of the objective we devise a Divide and Conquer method to evaluate significantly fewer values of Theta. This methods generalizes to binary linear robust problems. Specifically for shortest paths we derive a lower bound to speed-up the Divide and Conquer of Theta. The bound is based on carefully using previous shortest path computations. We combine the approach with non-preprocessing based acceleration techniques for Dijkstra adapted to the robust case. In a computational study we document the value of different accelerations tried in the algorithm engineering process. We also give an approximation scheme for the robust shortest path problem which computes a (1 + epsilon)-approximate solution requiring O(log(d^ / (1 + epsilon))) computations of the nominal problem where d^ := max d(A) / min (d(A){0})

Tree-Encoded Bitmaps

We propose a novel method to represent compressed bitmaps. Similarly to existing bitmap compression schemes, we exploit the compression potential of bitmaps populated with consecutive identical bits, i.e., 0-runs and 1-runs. But in contrast to prior work, our approach employs a binary tree structure to represent runs of various lengths. Leaf nodes in the upper tree levels thereby represent longer runs, and vice versa. The tree-based representation results in high compression ratios and enables efficient random access, which in turn allows for the fast intersection of bitmaps. Our experimental analysis with randomly generated bitmaps shows that our approach significantly improves over state-of-the-art compression techniques when bitmaps are dense and/or only barely clustered. Further, we evaluate our approach with real-world data sets, showing that our tree-encoded bitmaps can save up to one third of the space over existing techniques

Mobility-On-Demand Service In Mass Transit: Hypercommute options

Digitization, increasing automation and new business models like shared mobility have revolutionized transportation and mobility. Ridesharing companies like Uber and Lyft provide technological platforms and support to connect drivers and riders on the basis of demand-response services. Although the most improvements in on-demand applications have been experimented in private transit services, there is no any implementation in public transportation to connect public transit services and passengers each other. Ondemand is still vague. However, providing on-demand services in public transportation is complicated because of the big capacity problem in mass transit, its application in public transit services can enable flexible mobility for riders and provide personalized mobility experience. This paper presents the concept of mobility-on-demand service and its application in public transit services with an technological innovation of FM/LM pilot project represented by HyperCommute. The paper starts with introduction, then the business model of mobility-on-demand service is described and the most used algorithms are explained, then an illustrative example of HyperCommute mobility-on-demand service is given. Also, the applicability of mobility-on-demand service in Istanbul is discussed. The paper ends up with conclusion and future directions

Efficient Algorithms for Solving Size-Shape-Topology Truss Optimization and Shortest Path Problems

Efficient numerical algorithms for solving structural and Shortest Path (SP) problems are proposed and explained in this study. A variant of the Differential Evolution (DE) algorithm for optimal (minimum) design of 2-D and 3-D truss structures is proposed. This proposed DE algorithm can handle size-shape-topology structural optimization. The design variables can be mixed continuous, integer/or discrete values. Constraints are nodal displacement, element stresses and buckling limitations. For dynamic (time dependent) networks, two additional algorithms are also proposed in this study. A heuristic algorithm to find the departure time (at a specified source node) for a given (or specified) arrival time (at a specified destination node) of a given dynamic network. Finally, an efficient bidirectional Dijkstra shortest path (SP) heuristic algorithm is also proposed. Extensive numerical examples have been conducted in this study to validate the effectiveness and the robustness of the proposed three numerical algorithms

Shortest path algorithms for dynamic transportation networks

Over the last decade, many interesting route planning problems can be solved by finding the shortest path in a weighted graph that represents a transportation network. Such networks are private transport networks or timetabled public transportation networks. In the shortest path problem, every network type requires different algorithms to compute one or more than one shortest path. However, routing in a public transportation network is completely different and is much more complex than routing in a private transport network, and therefore different algorithms are required. For large networks, the standard shortest path algorithms - Dijkstra's algorithm (1959) and Bellman's algorithm (1958)- are too slow. Consequently, faster algorithms have been designed to speed up the search. However, these algorithms often consider only the simplest scenario of finding an optimal route on a graph with static real edge costs. But real map routing problems are often not that simple – it is often necessary to consider time-dependent edge costs. For example, in public transportation routing, consideration of the time-dependent model of these networks is mandatory. However, there are a number of transportation applications that use informed search algorithms (where the algorithm uses heuristics that guide the search toward the destination), rather than one of the standard static shortest path algorithms. This is primarily due to shortest paths needing to be rapidly identified either because an immediate response is required. For example, the A* algorithm (Nilsson, 1971) is widely used in artificial intelligence. Heuristic information (in the form of estimated distance to the destination) is used to focus the search towards the destination node. This results in finding the shortest path faster than the standard static search algorithms. Road traffic congestion has become an increasingly significant problem in a modern society. In a dynamic traffic environment, traffic conditions are time-dependent. For instance, when travelling from home to the work, although an optimal route can be planned prior to departure based on the traffic conditions at that time, it may be necessary to adjust the route while en route because traffic conditions change all the time. In some cases, it is necessary to modify the travelling route from time to time and re-plan a new route from the current location to the destination, based on the real-time traffic information. The challenge lies in the fact that any modification to the optimal route to adapt to the dynamic environment necessitates speeding up of the search efforts. Among the algorithms suggested for the dynamic shortest path problem is the algorithm of Lifelong Planning A* algorithm (LPA*) (Koenig, Likhachev and Furcy, 2004). This algorithm has been given this name because of its ability to reuse information from previous searches. It is used to adjust a shortest path to adapt to the dynamic transportation network. Search space and fast shortest path queries can be used for finding fastest updated route on road and bus networks. Consequently, the efficient processing of both types of queries is of first-rate significance. However, most search methods focus only on one type of query and do not efficiently support the other. To address this challenge, this research presents the first novel approach; an Optimised Lifelong Planning A* (OLPA*) algorithm. The OLPA* used an appropriate data structure to improve the efficiency of the dynamic algorithms implementation making it capable of improving the search performance of the algorithm to solve the dynamic shortest path problem, which is where the traveller may have to re-compute the shortest path while travelling in a dynamic transportation environment. This research has also proposed bi-directional LPA* (BLPA*) algorithm. The proposed algorithm BLPA* used bi-directional search strategy and the main idea in this strategy is to divide the search problem into two separate problems. One search proceeds forwards from the start node, while the other search proceeds backwards from the end node. The solution requires the two search problems to meet at one middle node. The BLPA* algorithm has the same overall structure as the LPA* algorithm search, with some differences that the BLPA* contains a priority queue for each direction. This research presented another algorithm that designed to adaptively derive the shortest path to the desired destination by making use of previous search results and reducing the total execution time by using the benefits of a bi-directional search strategy . This novel algorithm has been called the bi-directional optimised Lifelong A* algorithm (BiOLPA*). It was originally proposed for road transport networks and later also applied to public transportation networks. For the road transport network, the experimental results demonstrate that the proposed incremental search approach considerably outperforms the original approach method, which recomputed the shortest path from scratch each time without utilization of the previous search results. However, for public transportation, the significant problem is that it is not possible to apply a bi-directional search backwards using estimated arrival time. This has been further investigated and a better understanding of why this technique fails has been documented. While the OLPA* algorithms give an impressive result when applied on bus network compared with original A* algorithms, and our experimental results demonstrate that the BiOLPA* algorithm on road network is significantly faster than the LPA*, OLPA* and the A* algorithms, not only in terms of number of expansion nodes but also in terms of computation time