As a dual problem of influence maximization, the seed minimization problem asks for the minimum number of seed nodes to influence a required number eta of users in a given social network G. Existing algorithms for seed minimization mostly consider the it non-adaptive setting, where all seed nodes are selected in one batch without observing how they may influence other users. In this paper, we study seed minimization in the it adaptive setting, where the seed nodes are selected in several batches, such that the choice of a batch may exploit information about the actual influence of the previous batches. We propose a novel algorithm, it ASTI, which addresses the adaptive seed minimization problem in OBig(fraceta cdot (m+n)varepsilon^2ln n Big) expected time and offers an approximation guarantee of frac(ln eta+1)^2(1 - (1-1/b)^b) (1-1/e)(1-varepsilon) in expectation, where eta is the targeted number of influenced nodes, b is size of each seed node batch, and varepsilon in (0, 1) is a user-specified parameter. To the best of our knowledge, ASTI is the first algorithm that provides such an approximation guarantee without incurring prohibitive computation overhead. With extensive experiments on a variety of datasets, we demonstrate the effectiveness and efficiency of ASTI over competing methods.