## 1 Introduction  
The popularization of smartphones has made people increasingly connected, and has provided the great convenience for daily order delivery service. For example, by using the ride-ordering apps, such as DiDi Chuxing (www.didiglobal.com) and Uber (www.uber.com), customers can be picked up by the private car or taxi at any time, and, by using the food-ordering apps, such as Meituan (www.meituan.com), Eleme (www.ele.me), and Freshhema (www.freshhema.com), food can be delivered to customers' homes within a very short time. During July 2019, the average daily ride orders of DiDi Chuxing broke 24 million, and the average daily food orders of Meituan broke 30 million. Delivery service routing (**DSR**), which routes the cars/couriers to deliver orders, has made such an important contribution of serving these millions of orders. To improve customer experience, a significant characteristic of the ordering apps is to provide the flexible pickup or drop-off windows [1; 2]. For example, in the food-ordering apps, the customer has the flexibility to choose the food drop-off time, e.g., during the lunch time, it is feasible to drop off the food from 11:30 a.m. to 12:30 p.m. Moreover, to improve the system throughput (i.e., the number of orders served), multiple orders can be shared and served by the same car/courier at a time [3; 4; 5]. For example, in the food-ordering scenarios, this food ordered from the same shop can be packed by a courier and delivered to the customers in a sharing fashion. Due to its potential in real-world applications, this paper mainly focuses on delivery service sharing variant with flexible pickup/drop-off time windows (**DSS-Fle**), where orders can be shared and served at a time. Recently, many efforts have been devoted to addressing two variants of DSS problem-DSR with flexible time windows (DSR-Fle) and DSS with a fixed time window (DSS-Fix). In the DSR-Fle literature, where orders cannot be shared, [PERSON] et al. [6] first propose a shareability network to model the special DSR variant with fixed pickup window (**DSR-Fix**), which can be addressed by the polynomial Hopcroft-Karp maximum-matching algorithm. [PERSON] et al. [7] extend to the NP-hard DSR-Fle, and propose an iterative heuristic algorithm of removing the undesirable edges in the shareability network in each iteration. Although the heuristic algorithm can scale to real-world city-scale scenarios, it has no theoretical guarantee on system throughput. On the other hand, a mixed integer program is used to formulate the DSS-Fix variant, and the time-consuming Lagrangian decomposition approach is proposed for the exact solution [8]. To reduce the computation complexity, a bipartite matching based approximation algorithm is proposed for matching a delivery resource with two orders [9], which might be inefficient for system throughput maximization. [PERSON] et al. [10] propose a heuristic algorithm to generate the order sharing path, which works by finding and inserting the nearby orders into a current partial sharing path. The constructed sharing path might be efficient on system throughput and computation time; however, neither considers the flexible time windows nor provides any systemic analysis on the advantage of order sharing. Against this background, the main contribution of this paper is to design a time-efficient algorithm for DSS-Fle, and to theoretically guarantee the advantage of order sharing. We extend the seminal shareability network mechanism (that is used in DSR-Fix), to DSR-Fle and DSS-Fix, respectively. For DSR-Fle, we split the flexible time windows into multiple virtual fixed times, carefully choosing one of these fixed times as the order's service time, and employ the DSR-Fix mechanism to generate the routing plan. The DSR-Fle variant algorithm can provide the theoretical \(\frac{1}{x}\) approximation ratio with respect to system throughput, where \(\kappa\) is the maximum time window of orders.