We propose highly efficient heuristic and exact algorithms to solve the Electric Autonomous Dial-A-Ride Problem (E-ADARP), which consists in designing a set of minimum-cost routes that accommodates all customer requests for a fleet of Electric Autonomous Vehicles (EAVs). The E-ADARP has two important features: (i) the employment of EAVs and a partial recharging policy; (ii) the weighted-sum objective function that minimizes the total travel time and the total excess user ride time. We first propose a Deterministic Annealing (DA) algorithm to solve the E-ADARP. Partial recharging (i) is handled by an exact route evaluation scheme of linear time complexity. To tackle (ii), we propose a new method that allows effective computations of minimum excess user ride time by introducing a fragment-based representation of paths. To validate the performance of the DA algorithm, we compare our algorithm results to the best-reported Branch-and-Cut (B&C) algorithm results on existing instances. Our DA algorithm provides 25 new best solutions and 45 equal solutions for 84 existing instances. To test the algorithm’s performance on larger-sized instances, we establish new instances with up to 8 vehicles and 96 requests, and we provide 19 new solutions for these instances. Then, we present a highly efficient CG algorithm, which is integrated into the Branch-and-price (B&P) scheme to solve the E-ADARP exactly. Our CG algorithm relies on an effective labeling algorithm to generate columns with negative reduced costs. In the extension of labels, the key challenge is determining all excess-user-ride-time optimal schedules to ensure finding the minimum-negative-reduced-cost route. To handle this issue, we apply the fragment-based representation and propose a novel approach to abstract fragments to arcs while ensuring excess-user-ride-time optimality. We then construct a new graph that preserves all feasible routes of the original graph by enumerating all feasible fragments, abstracting them to arcs, and connecting them with each other, depots, and recharging stations in a feasible way. On the new graph, we apply strong dominance rules and constant-time feasibility checks to compute the shortest paths efficiently. In the computational experiments, we solve 71 out of 84 instances optimally, improve 30 previously reported lower bounds, and generate 41 new best solutions on previously solved and unsolved instances.