Optimal rules for the sequential selection of monotone subsequences of maximum expected length

This article presents new results on the problem of selecting (online) a monotone subsequence of maximum expected length from a sequence of i.i.d. random variables. We study the case where the variables are observed sequentially at the occurrence times of a Poisson process with known rate. Our approach is a detailed study of the integral equation which determines v(t), the expected number (under the optimal strategy for time horizon t) of selected points Ltt up to time t. We first show that v(t), v′(t) and v″(t) exist everywhere on R+. Then, in particular, we prove that v″(t)<0 for all t∈[0,∞[, implying that v is strictly concave on R+. This settles a conjecture of Gnedin and opens the way to stronger bounds for v and its derivatives. We can show v′(t)2t∼1 from which we derive new and much tighter bounds for v(t), namely2t−log(1+2t)+c⩽v(t)⩽2t.Using a martingale approach, we can show that the variance of Ltt satisfies13v(t)⩽Var(Ltt)⩽13v(t)+c1 log(t)+c2.Further we obtain several results on the process (Lut)0⩽u⩽t, where Lut denotes the number of selected points up to time u when applying the optimal rule with respect to the time horizon t. We will also show that the integral equation which yields v(t), has a unique solution. As an application, this result is used to extend a known result on the equivalence of a specific bin-packing problem with a certain subsequence problem. Our more personal interest in quick selection rules and their performance leads us also to the study of a class of convenient graph-rules. Known results on the concentration measure of record values suggest that the asymptotically best graph-rule should be the diagonal line rule, and we prove this intuition to be correct.