Optimal online selection of a monotone subsequence: a central limit theorem

Abstract

Abstract. Consider a sequence of n independent random variables with a common continuous distribution F , and consider the task of choosing an increasing subsequence where the observations are revealed sequentially and where an observation must be accepted or rejected when it is first revealed. There is a unique selection policy π * n that is optimal in the sense that it maximizes the expected value of Ln(π * n ), the number of selected observations. We investigate the distribution of Ln(π * n ); in particular, we obtain a central limit theorem for Ln(π * n ) and a detailed understanding of its mean and variance for large n. Our results and methods are complementary to the work of Bruss and Delbaen (2004) where an analogous central limit theorem is found for monotone increasing selections from a finite sequence with cardinality N where N is a Poisson random variable that is independent of the sequence.

An adaptive O(log n)‐optimal policy for the online selection of a monotone subsequence from a random sample

“…After a careful analysis, Bruss and Delbaen proved that the mean‐optimal number of monotone increasing selections with Poisson‐many observations is asymptotically normal after centering around and scaling by . Arlotto et al showed that the same asymptotic limit also holds for the discrete‐time problem with n observations so, in summary, we now know that …”

“…, n-the sequential nature of the two selection processes makes the result of Bruss and Delbaen [5] not immediately applicable. The connection between the continuous-time formulation of Bruss and Delbaen [5] and the discrete-time optimization (2) was then argued by Arlotto et al [2] who used the concavity of the map n → E[L n (π * n )] and the O(log n)-bound of Bruss and Delbaen [5] to ultimately confirm the lower bound in (4).…”

“…so we have that the optimal objective value of (13) is an upper bound for E[L k (π k , s)], and that (13) is a relaxation of the online monotone subsequence problem (2). The optimal value of (13) can be easily estimated.…”

“…While the Bruss and Delbaen result provides compelling evidence that a similar bound should also hold for the discrete‐time formulation we consider here—that is the formulation in which the observations are revealed at the times —the sequential nature of the two selection processes makes the result of Bruss and Delbaen not immediately applicable. The connection between the continuous‐time formulation of Bruss and Delbaen and the discrete‐time optimization (2) was then argued by Arlotto et al who used the concavity of the map and the ‐bound of Bruss and Delbaen to ultimately confirm the lower bound in (4).…”

Quickest online selection of an increasing subsequence of specified size

“…In Sections 5 and 6, we develop the relationship between Theorems 1 and 2 and the more traditional size-focused online selection problem which was first studied in Samuels and Steele [15] and then studied much more extensively by Bruss and Robertson [9], Gnedin [11], Bruss and Delbaen [7,8], and Arlotto et al [3]. On an intuitive level, the time-focused selection problem and the size-focused selection problems are dual to each other, and it is curious to consider the extent to which rigorous relationships that can be developed between the two.…”