TY - JOUR

T1 - Hiring Secretaries over Time

T2 - The Benefit of Concurrent Employment

AU - Disser, Yann

AU - Fearnley, John

AU - Gairing, Martin

AU - Göbel, Oliver

AU - Klimm, Max

AU - Schmand, Daniel

AU - Skopalik, Alexander

AU - Tönnis, Andreas

PY - 2020/2

Y1 - 2020/2

N2 - We consider a stochastic online problem where n applicants arrive over time, one per time step. Upon the 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; that is, 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 2.965 for the case of uniform distributions. For this case, we give an analytical lower bound of 2 and a computational lower bound of 2.148. We then adapt our algorithm to stay competitive even in settings with one or more of the following restrictions: (i) at most two applicants can be hired concurrently; (ii) the distribution of the applicants’ costs is unknown; (iii) the total number n of time steps is unknown. On the other hand, we show that concurrent employment is a necessary feature of competitive algorithms by proving that no algorithm has a competitive ratio better than Ω(√n/log n) if concurrent employment is forbidden.

AB - We consider a stochastic online problem where n applicants arrive over time, one per time step. Upon the 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; that is, 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 2.965 for the case of uniform distributions. For this case, we give an analytical lower bound of 2 and a computational lower bound of 2.148. We then adapt our algorithm to stay competitive even in settings with one or more of the following restrictions: (i) at most two applicants can be hired concurrently; (ii) the distribution of the applicants’ costs is unknown; (iii) the total number n of time steps is unknown. On the other hand, we show that concurrent employment is a necessary feature of competitive algorithms by proving that no algorithm has a competitive ratio better than Ω(√n/log n) if concurrent employment is forbidden.

KW - Markov chain

KW - Online algorithm

KW - Prophet inequality

KW - Secretary problem

KW - Stopping problem

KW - 22/2 OA procedure

UR - http://www.scopus.com/inward/record.url?scp=85084757364&partnerID=8YFLogxK

U2 - 10.1287/moor.2019.0993

DO - 10.1287/moor.2019.0993

M3 - Article

VL - 45

SP - 323

EP - 352

JO - Mathematics of operations research

JF - Mathematics of operations research

SN - 0364-765X

IS - 1

ER -