We study exact, efficient and practical algorithms for route planning in large road networks. Routing applications often require integrating the current traffic situation, planning ahead with traffic predictions for the future, respecting forbidden turns, and many other features depending on the exact application. While Dijkstra's algorithm can be used to solve these problems, it is too slow for many applications. A* is a classical approach to accelerate Dijkstra's algorithm. A* can support many extended scenarios without much additional implementation complexity. However, A*'s performance depends on the availability of a good heuristic that estimates distances. Computing tight distance estimates is a challenge on its own. On road networks, shortest paths can also be quickly computed using hierarchical speedup techniques. They achieve speed and exactness but sacrifice A*'s flexibility. Extending them to certain practical applications can be hard. In this paper, we present an algorithm to efficiently extract distance estimates for A* from Contraction Hierarchies (CH), a hierarchical technique. We call our heuristic CH-Potentials. Our approach allows decoupling the supported extensions from the hierarchical speed-up technique. Additionally, we describe A* optimizations to accelerate the processing of low degree nodes, which often occur in road networks.

We study the problem of computing shortest paths in massive road networks with traffic predictions. Incorporating traffic predictions into routing allows, for example, to avoid commuter traffic congestions. Existing techniques follow a two-phase approach: In a preprocessing step, an index is built. The index depends on the road network and the traffic patterns but not on the path start and end. The latter are the input of the query phase, in which shortest paths are computed. All existing techniques have either large index size, slow query running times, or may compute suboptimal paths. In this work, we introduce CATCHUp (Customizable Approximated Time-dependent Contraction Hierarchies through Unpacking), the first algorithm that simultaneously achieves all three objectives. The core idea of CATCHUp is to store paths instead of travel times at shortcuts. Shortcut travel times are derived lazily from the stored paths. We perform an experimental study on a set of real world instances and compare our approach with state-of-the-art techniques. Our approach achieves the fastest preprocessing, competitive query running times and up to 30 times smaller indexes than competing approaches.

The notion of treewidth, introduced by Robertson and Seymour in their seminal Graph Minors series, turned out to have tremendous impact on graph algorithmics. Many hard computational problems on graphs turn out to be efficiently solvable in graphs of bounded treewidth: graphs that can be sweeped with separators of bounded size. These efficient algorithms usually follow the dynamic programming paradigm. In the recent years, we have seen a rapid and quite unexpected development of involved techniques for solving various computational problems in graphs of bounded treewidth. One of the most surprising directions is the development of algorithms for connectivity problems that have only single-exponential dependency (i.e., 2^{{O}(t)}) on the treewidth in the running time bound, as opposed to slightly superexponential (i.e., 2^{{O}(t log t)}) stemming from more naive approaches. In this work, we perform a thorough experimental evaluation of these approaches in the context of one of the most classic connectivity problem, namely Hamiltonian Cycle.

Quasi-threshold graphs are $\{C_4, P_4\}$-free graphs, i.e., they do not contain any cycle or path of four nodes as an induced subgraph. We study the $\{C_4, P_4\}$-free editing problem, which is the problem of finding a minimum number of edge insertions or deletions to transform an input graph into a quasi-threshold graph. This problem is NP-hard but fixed-parameter tractable (FPT) in the number of edits by using a branch-and-bound algorithm and admits a simple integer linear programming formulation (ILP). Both methods are also applicable to the general $F$-free editing problem for any finite set of graphs $F$. For the FPT algorithm, we introduce a fast heuristic for computing high-quality lower bounds and an improved branching strategy. For the ILP, we engineer several variants of row generation. We evaluate both methods for quasi-threshold editing on a large set of protein similarity graphs. For most instances, our optimizations speed up the FPT algorithm by one to three orders of magnitude. The running time of the ILP, that we solve using Gurobi, becomes only slightly faster. With all optimizations, the FPT algorithm is slightly faster than the ILP, even when listing all solutions. Additionally, we show that for almost all graphs, solutions of the previously proposed quasi-threshold editing heuristic QTM are close to optimal.

We study the earliest arrival and profile problems in road networks with time-dependent functions as arc weights and dynamic updates. We present and experimentally evaluate simple, sampling-based, heuristic algorithms. Our evaluation is performed on large, current, production-grade road graph data with time-dependent arc weights. It clearly shows that the proposed algorithms are fast and compute paths with a sufficiently small error for most practical applications. We experimentally compare our algorithm against the current state-of-the-art. Our experiments reveal, that the memory consumption of existing algorithms is prohibitive on large instances. Our approach does not suffer from this limitation. Further, our algorithm is the only competitor able to answer profile queries on all test instances below 50ms. As our algorithm is simple to implement, we believe that it is a good fit for many realworld applications.

We study the problem of computing traffic assignments for public transit networks: Given a public transit network and a demand (i.e. a list of passengers, each with associated origin, destination, and departure time), the objective is to compute the utilization of every vehicle. Efficient assignment algorithms are a core component of many urban traffic planning tools. In this work, we present a novel algorithm for computing public transit assignments. Our approach is based upon a microscopic Monte Carlo simulation of individual passengers. In order to model realistic passenger behavior, we base all routing decisions on travel time, number of transfers, time spent walking or waiting, and delay robustness. We show how several passengers can be processed during a single scan of the network, based on the Connection Scan Algorithm [Dibbelt et al., LNCS Springer 2013], resulting in a highly efficient algorithm. We conclude with an experimental study, showing that our assignments are comparable in terms of quality to the state-of-the-art. Using the parallelized version of our algorithm, we are able to compute a traffic assignment for more than ten million passengers in well below a minute, which outperforms previous works by more than an order of magnitude.

We introduce FlowCutter, a novel algorithm to compute a set of edge cuts or node separators that optimize cut size and balance in the Pareto-sense. Our core algorithm heuristically solves the balanced connected st-edge-cut problem, where two given nodes s and t must be separated by removing edges to obtain two connected parts. Using the core algorithm we build variants that compute node separators and are independent of s and t. Using the computed Pareto-set we can identify cuts with a particularly good trade-off between cut size and balance that can be used to compute contraction and minimum fill-in orders, which can be used in Customizable Contraction Hierarchies (CCH), a speedup technique for shortest path computations. Our core algorithm runs in O(cm) time where m is the number of edges and c the cut size. This makes it well-suited for large graphs with small cuts, such as road graphs, which are our primary application. For road graphs we present an extensive experimental study demonstrating that FlowCutter outperforms the current state of the art both in terms of cut sizes as well as CCH performance.

In this work we give a first tractability analysis of Compressed Path Databases, space efficient oracles used to very quickly identify the first arc on a shortest path. We study the complexity of computing an optimal compressed path database for general directed and undirected graphs. We find that in both cases the problem is NP-complete. We also show that, for graphs which can be decomposed along articulalion points, the problem can be decomposed into independent parts, with a corresponding reduction in its level of difficulty. In particular, this leads to simple and tractable algorithms which yield optimal compression results for trees.

We study a scenario for route planning in road networks, where the objective to be optimized may change between every shortest path query. Since this invalidates many of the known speedup techniques for road networks that are based on preprocessing of shortest path structures, we investigate optimizations exploiting topological structures. We experimentally evaluate our technique on a large set of real-world road networks of various data sources. With lightweight preprocessing our technique answers long-distance queries across continental networks significantly faster than previous approaches towards the same problem formulation.

The Grid-Based Path Planning Competition has just completed its third iteration. The entriesused in the competition have improved significantly during this time, changing the view ofthe state of the art of grid-based pathfinding. Furthermore, the entries from the competition have beenmade publicly available, improving the ability of researchers to compare their work. Thispaper summarizes the entries to the 2014 competition, presents the 2014 competition results,and talks about what has been learned and where there is room for improvement.

We introduce a novel approach to Compressed Path Databases, space efficient oracles used to very quickly identify the first edge on a shortest path. Our algorithm achieves query running times on the 100 nanosecond scale, being significantly faster than state-of-the-art first-move oracles from the literature. Space consumption is competitive, due to a compression approach that rearranges rows and columns in a first-move matrix and then performs run length encoding (RLE) on the contents of the matrix. One variant of our implemented system was, by a convincing margin, the fastest entry in the 2014 Grid-Based Path Planning Competition. We give a first tractability analysis for the compression scheme used by our algorithm. We study the complexity of computing a database of minimum size for general directed and undirected graphs. We find that in both cases the problem is NP-complete. We also show that, for graphs which can be decomposed along articulation points, the problem can be decomposed into independent parts, with a corresponding reduction in its level of difficulty. In particular, this leads to simple and tractable algorithms with linear running time which yield optimal compression results for trees.

We study the problem of efficiently computing journeys in timetable networks. Our algorithm optimally answers profile queries, computing all journeys given a time interval. Our study demonstrates that queries can be answered optimally on large country-scale timetable networks within several milliseconds and fast delay integration is possible. Previous work either had to drop optimality or only considered comparatively small timetable networks. Our technique is a combination of the Connection Scan Algorithm and multilevel overlay graphs.

We study the problem of computing delay-robust routes in timetable networks. Instead of a single path we compute a decision graph containing all stops and trains/vehicles that might be relevant. Delays are formalized using a stochastic model. We show how to compute a decision graph that minimizes the expected arrival time while bounding the latest arrival time over all sub-paths. Finally we show how the information contained within a decision graph can compactly be represented to the user. We experimentally evaluate our algorithms and show that the running times allow for interactive usage on a realistic train network.

We introduce a novel preprocessing-based algorithm to solve the problem of determining the first arc of a shortest path in sparse graphs. Our algorithm achieves query running times on the 100 nanosecond scale, being significantly faster than state-of-the-art first-move oracles from the literature. Space consumption is competitive, due to a compression approach that rearranges rows and columns in a first-move matrix and then performs run length encoding (RLE) on the contents of the matrix.