Abstract: Imagine a large firm with multiple departments that plans a large recruitment. Candidates arrive one-by-one, and for each candidate the firm decides, based on her data (CV, skills, experience, etc), whether to summon her for an interview. The firm wants to recruit the best candidates while minimizing the number of interviews. We model such scenarios as an assignment problem between items (candidates) and categories (departments): the items arrive one-by-one in an online manner, and upon processing each item the algorithm decides, based on its value and the categories it can be matched with, whether to retain or discard it (this decision is irrevocable). The goal is to retain as few items as possible while guaranteeing that the set of retained items contains an optimal matching. We consider two variants of this problem: (i) in the first variant it is assumed that the $n$ items are drawn independently from an unknown distribution $D$. (ii) In the second variant it is assumed that before the process starts, the algorithm has an access to a training set of $n$ items drawn independently from the same unknown distribution (e.g.\ data of candidates from previous recruitment seasons). We give tight bounds on the minimum possible number of retained items in each of these variants. These results demonstrate that one can retain exponentially less items in the second variant (with the training set). Our algorithms and analysis utilize ideas and techniques from statistical learning theory and from discrete algorithms.