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Ride-hailing sharing involves grouping ride-hailing customers with similar trips and time schedules to share the same ride-hailing vehicle to reduce their total travel cost. With the current information and communication technology, ride-hailing customers and drivers can be matched in real-time via a ride-hailing platform. This paper formulates a dynamic ride-hailing sharing problem that simultaneously maximizes the number of served customers, minimizes the travel cost and travel time ratios, and considers the capacity, time window, and travel cost constraints. The travel cost ratio is the ratio of actual passengers’ fare to the passengers’ fare without ride-hailing sharing, whereas the travel time ratio is defined as the actual travel time (including waiting time) over the maximum allowable travel time. To solve the dynamic problem, it is divided into many small and continuous static subproblems with an equal time interval. Each subproblem is solved by a modified artificial bee colony (MABC) algorithm with path relinking, while the contraction hierarchies and vantage point tree are used to determine the shortest path and accelerate the algorithm, respectively. Problem properties and the performance of the proposed solution method are demonstrated using large-scale real-time data from Didi that is the largest ride-hailing company in China. The proposed method is shown to outperform the benchmark, i.e., greedy randomized adaptive search procedure (GRASP) with path relinking. The proposed method also performs better when the length of each time interval is longer, and the tolerance for the incremental travel time caused by detours is higher. We also demonstrate that (a) considering both travel cost and travel time ratios in the objective can achieve a better sharing percentage, and balance the increase in the travel time ratio and the decrease in the travel cost ratio compared with the objective that misses either travel time or the travel cost ratio; and (b) the passengers can gain a large out-of-pocket cost saving in the case of ride-hailing sharing while enduring a relatively small increase in travel time compared with the case without ride-hailing sharing.

Contraction hierarchies are a speed-up technique to improve the performance of shortest-path computations, which works very well in practice. Despite convincing practical results, there is still a lack of theoretical explanation for this behavior.In this paper, we develop a theoretical framework for studying search space sizes in contraction hierarchies. We prove the first bounds on the size of search spaces that depend solely on structural parameters of the input graph, that is, they are independent of the edge lengths. To achieve this, we establish a connection with the well-studied elimination game. Our bounds apply to graphs with treewidth k, and to any minor-closed class of graphs that admits small separators. For trees, we show that the maximum search space size can be minimized efficiently, and the average size can be approximated efficiently within a factor of 2.We show that, under a worst-case assumption on the edge lengths, our bounds are comparable to those in the recent paper "VC-Dimension and Shortest Path Algorithms" of Abraham et al. 1, whose analysis depends also on the edge lengths. As a side result, we link their notion of highway dimension (a parameter that is conjectured to be small, but is unknown for all practical instances) with the notion of pathwidth. This is the first relation of highway dimension with a well-known graph parameter.

We present a novel algorithm to solve the nonnegative single-source shortest path problem on road networks and other graphs with low highway dimension. After a quick preprocessing phase, we can compute all distances from a given source in the graph with essentially a linear sweep over all vertices. Because this sweep is independent of the source, we are able to reorder vertices in advance to exploit locality. Moreover, our algorithm takes advantage of features of modern CPU architectures, such as SSE and multi-core. Compared to Dijkstra's algorithm, our method needs fewer operations, has better locality, and is better able to exploit parallelism at multi-core and instruction levels. We gain additional speedup when implementing our algorithm on a GPU, where our algorithm is up to three orders of magnitude faster than Dijkstra's algorithm on a high-end CPU. This makes applications based on all-pairs shortest-paths practical for continental-sized road networks. Several algorithms, such as computing the graph diameter, exact arc flags, or centrality measures (exact reaches or betweenness), can be greatly accelerated by our method.