New results for the k-secretary problem

Abstract Suppose that n items arrive online in random order and the goal is to select k of them such expected sum selected maximized. The decision for any item irrevocable must be made on arrival without knowing future items. This problem known as k-secretary problem, which includes classical secretary with special case = 1 . It well-known latter can solved by a simple algorithm competitive ratio / e optimal ? ? Existing algorithms beating threshold either rely involved selection policies already 2 , or assume large. In this paper we present results considering interesting relevant small. We focus algorithms, accompanied combinatorial analyses. As main contribution propose natural deterministic designed have ratios strictly greater than small ? hardly more complex elegant strategy works all derive its ? 100 ranging from 0.41 0.75 Moreover, consider an proposed earlier literature, no rigorous analysis known. show 0.4168 implying previous was not tight. Our reveals surprising property algorithm, might helpful find tight k.