ަ௨ ˒ ಓ࿏ωοτϫʔΫʴ߱٬ ˒ ӡߦ࿏ઢʴӺɾۭߓͷ߱٬ ˒ ܈ɾө૾ ˒ ΦϒδΣΫτͷܗঢ়ʴ࣍ݩ࠲ඪ ˒ ྠֲɾςΫενϟʴըૉ 5 ෳࡶͳߏΛ࣋ͭେنσʔλͷวࡏ จͷඃҾ༻ωοτϫʔΫ ϚϯϋολϯͷλΫγʔ߱٬σʔλ COMPUTER SCIENCES SOCIAL SCIENCES likely to be required in the future. A schema of the method is shown in Fig. 1. Our formulation is flexible with respect to physical and performance-related constraints that might need to be added. In our implementation, we consider the following. (i) For each request r, the waiting time !r , given by the difference between the pickup time tp r and the request time tr r , must be below a max- imum waiting time ⌦, for example, 2 min. (ii) For each passenger or request r the total travel delay r = td r t⇤ r must be lower than a maximum travel delay , for example, 4 min, where td r is the drop-off time and t⇤ r = tr r +⌧(or , dr ) is the earliest possible time at which the destination could be reached if the shortest path between the origin or and the destination dr was followed with- out any waiting time. The total travel delay r includes both the in-vehicle delay and the waiting time. Finally, (iii) for each vehi- cle v, we consider a maximum number of passengers, npass v ⌫, for example, capacity 10. We define the cost C of an assignment as the sum of delays r (which includes the waiting time) over all assigned requests and passengers, plus a large constant cko for each unassigned request. Given an assignment ⌃ of requests to vehicles, we denote by Rok the set of requests that have been assigned to some vehicle and Rko the set of unassigned requests, due to the constraints or the fleet size. Formally, C(⌃) = X v2V X r2Pv r + X r2Rok r + X r2Rko cko . [1] This constrained optimization problem is solved via four steps (Fig. 1), which are: computing a pairwise request-vehicle share- ability graph (RV-graph) (Fig. 1B); computing a graph of fea- sible trips and the vehicles that can serve them (RTV-graph) (Fig. 1C); solving an ILP to compute the best assignment of vehi- cles to trips (Fig. 1D); and rebalancing the remaining idle vehi- cles (Fig. 1E). Given a network graph with travel times, we consider a func- tion travel(v, Rv ) for single-vehicle routing. For a vehicle v, with passengers Pv , this function returns the optimal travel route v to satisfy requests Rv . This route minimizes the sum of delays P r2Pv [Rv r subject to the constraints Z (waiting time, delay, and capacity). For low-capacity vehicles, such as taxis, the optimal path can be computed via an exhaustive search. For vehicles with larger capacity, heuristic methods such as Lin–Kernighan (20), Tabu search (21), or simulated annealing (22) may be used. Fig. 2, Right shows the optimal route for a vehicle with four pas- sengers and an additional request. The RV-graph (Fig. 1B) represents which requests and vehi- cles might be pairwise-shared and builds on the idea of share- Fig. 2. (A) Snapshot: 2,000 vehicles, capacity of 4 (⌦ = 5 min, Wednesday, 2000 hours). Vehicle in the fleet are represented at their current positions. Colors indicate number of passengers (0: light blue; 1: light green; 2: yellow; 3: dark orange; 4: dark red); 39 rebalancing vehicles are displayed in dark blue—mostly in the upper Manhattan returning to the middle. (B) Close view of the scheduled path for a vehicle (dark red circle) with four passen- gers, which drops one off, picks up a new one (blue star), and drops all four. Drop-off locations are displayed with inverted triangles. See Movie S1 for a complete simulation. two types of edges: (i) edges e(r, T), between a request r and a trip T that contains request r (i.e., 9 e(r, T) , r 2 T), and (ii) edges e(T, v), between a trip T and a vehicle v that can exe- cute the trip (i.e., 9 e(T, v) , travel(v, T) is feasible). The cost P r2Pv [T r , sum of delays, is associated to each edge e(T,v). The algorithm to compute the feasible trips and edges pro- ceeds incrementally in trip size for each vehicle, starting from the request-vehicle edges in the RV-graph (SI Appendix, Algorithm 1). For computational efficiency, we rely on the fact that a trip T only needs to be checked for feasibility if there exists a vehicle v for which all of its subtrips T0 = T \ r (obtained by removing one request) are feasible and have been added as edges e(T0, v) to the RTV-graph. Next, we compute the optimal assignment ⌃optim of vehicles to trips. This optimization is formalized as an ILP, initialized with a greedy assignment obtained directly from the RTV-graph. To compute the greedy assignment ⌃ , trips are assigned to likely to be required in the future. A schema of the method is shown in Fig. 1. Our formulation is flexible with respect to physical and performance-related constraints that might need to be added. In our implementation, we consider the following. (i) For each request r, the waiting time !r , given by the difference between the pickup time tp r and the request time tr r , must be below a max- imum waiting time ⌦, for example, 2 min. (ii) For each passenger or request r the total travel delay r = td r t⇤ r must be lower than a maximum travel delay , for example, 4 min, where td r is the drop-off time and t⇤ r = tr r +⌧(or , dr ) is the earliest possible time at which the destination could be reached if the shortest path between the origin or and the destination dr was followed with- out any waiting time. The total travel delay r includes both the in-vehicle delay and the waiting time. Finally, (iii) for each vehi- cle v, we consider a maximum number of passengers, npass v ⌫, for example, capacity 10. We define the cost C of an assignment as the sum of delays r (which includes the waiting time) over all assigned requests and passengers, plus a large constant cko for each unassigned request. Given an assignment ⌃ of requests to vehicles, we denote by Rok the set of requests that have been assigned to some vehicle and Rko the set of unassigned requests, due to the constraints or the fleet size. Formally, C(⌃) = X v2V X r2Pv r + X r2Rok r + X r2Rko cko . [1] This constrained optimization problem is solved via four steps Fig. 2. (A) Snapshot: 2,000 vehicles, capacity of 4 (⌦ = 2000 hours). Vehicle in the fleet are represented at the Colors indicate number of passengers (0: light blue; 1: li 3: dark orange; 4: dark red); 39 rebalancing vehicles a blue—mostly in the upper Manhattan returning to th view of the scheduled path for a vehicle (dark red circl gers, which drops one off, picks up a new one (blue star Drop-off locations are displayed with inverted triangles complete simulation. IEEE SIG PROC MAG [IN7] 3D shape correspondence application: Finding intrin- sic correspondence between deformable shapes is a classical tough problem that underlies a broad range of vision and graphics applications, including texture mapping, animation, editing, and scene understanding [107]. From the machine learning standpoint, correspondence can be thought of as a classification problem, where each point on the query shape is assigned to one of the points on a reference shape (serving as a “label space”) [108]. It is possible to learn the correspondence with a deep intrinsic network applied to some input feature vector f(x) at each point x of the query shape X, producing an output U⇥(f(x))(y), which is interpreted as the conditional probability p(y|x) of x being mapped to y. Using a training set of points with their ground-truth correspondence {xi , yi}i2I, supervised learning is performed minimizing the multinomial regression loss min ⇥ X i2I log U⇥(f(xi))(yi) (64) w.r.t. the network parameters ⇥. The loss penalizes for the de- viation of the predicted correspondence from the groundtruth. We note that, while producing impressive result, such an approach essentially learns point-wise correspondence, which then has to be post-processed in order to satisfy certain properties such as smoothness or bijectivity. Correspondence is an example of structured output, where the output of the network at o (in the simp i.e., the out et al. [109] corresponden corresponden [FIGS7a] Lea U⇥ is applied The output of probability di of as a soft c [FIGS7b] Intrinsic correspondence established between human shapes using intrinsic layers). SHOT descriptors capturing the local normal vector orientations [110] were use is visualized by transferring texture from the leftmost reference shape. For additiona has brought a breakthrough in performance and led to an overwhelming trend in the community to favor deep learning formulation of but the mode ༷ʑͳϙʔζͷਓମܗঢ়σʔλ [Monti+ CVPR 2017] [Alonso-Mora+ PNAS 2017] [Bronstein+ IEEE SPM 2018]
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