Active search is an active learning setting with the goal of identifying as many members of a given class as possible under a labeling budget. In this work, we first establish a theoretical hardness of active search, proving that no polynomial-time policy can achieve a constant factor approximation ratio with respect to the expected utility of the optimal policy. We also propose a novel, computationally efficient active search policy achieving exceptional performance on several real-world tasks. Our policy is nonmyopic, always considering the entire remaining search budget. It also automatically and dynamically balances exploration and exploitation consistent with the remaining budget, without relying on a parameter to control this tradeoff. We conduct experiments on diverse datasets from several domains: drug discovery, materials science, and a citation network. Our efficient nonmyopic policy recovers significantly more valuable points with the same budget than several alternatives from the literature, including myopic approximations to the optimal policy.