Secretary Problems via Linear Programming

In the classical secretary problem an employer would like to choose the best candidate among n competing candidates that arrive in a random order. In each iteration, one candidate's rank vis-a-vis previously arrived candidates is revealed and the employer makes an irrevocable decision about her selection. This basic concept of n elements arriving in a random order and irrevocable decisions made by an algorithm have been explored extensively over the years, and used for modeling the behavior of many processes. Our main contribution is a new linear programming technique that we introduce as a tool for obtaining and analyzing algorithms for the secretary problem and its variants. The linear program is formulated using judiciously chosen variables and constraints and we show a one-to-one correspondence between algorithms for the secretary problem and feasible solutions to the linear program. Capturing the set of algorithms as a linear polytope holds the following immediate advantages: Computing the optimal algorithm reduces to solving a linear program. Proving an upper bound on the performance of any algorithm reduces to finding a feasible solution to the dual program. Exploring variants of the problem is as simple as adding new constraints, or manipulating the objective function of the linear program. We demonstrate these ideas by exploring some natural variants of the secretary problem. In particular, using our approach, we design optimal secretary algorithms in which the probability of selecting a candidate at any position is equal. We refer to such algorithms as position independent and these algorithms are motivated by the recent applications of secretary problems to online auctions. We also show a family of linear programs that characterize all algorithms that are allowed to choose J candidates and gain profit from the K best candidates. We believe that a linear programming based approach may be very helpful in the context of other variants of the secretary problem.