Hiring Secretaries over Time: The Benefit of Concurrent Employment

We consider a variant of the stochastic secretary problem where applicants need to be hired over time. Upon arrival of each applicant their cost per time step is revealed, and we have to fix the duration of employment, starting immediately. This decision is irrevocable, i.e., we can neither extend a contract nor dismiss a candidate once hired. In every time step, at least one candidate needs to be under contract, and our goal is to minimize the total hiring cost, which is the sum of the applicants’ costs multiplied with their respective employment durations. We provide a competitive online algorithm for the case that the applicants’ costs are drawn independently from a known distribution. Specifically, the algorithm achieves a competitive ratio of 2.98 for the case of uniform distributions, and we give a computational lower bound of 2.14. We are able to adapt our algorithm to stay competitive in settings where at most two applicants can be hired concurrently and/or the distribution of the applicants’ costs is unknown. On the other hand, we show that concurrent employment is a necessary feature of competitive algorithms by showing that no algorithm has a competitive ratio better than Ω( √ n/ logn) if concurrent employment is forbidden, even for uniform distributions and when the total number of time steps is known. 1998 ACM Subject Classification F.1.2 Modes of Computation,F.2 Analysis of Algorithms and Problem Complexity, G.1.6 Optimization