Sequential Selection of a Monotone Subsequence from a Random Permutation

We find a two term asymptotic expansion for the optimal expected value of a sequentially selected monotone subsequence from a random permutation of length n. A striking feature of this expansion is that it tells us that the expected value of optimal selection from a random permutation is quantifiably larger than optimal sequential selection from an independent sequence of uniformly distributed random variables; specifically, it is larger by at least (1/6) logn+O(1). 1. Sequential subsequence problems In the classical monotone subsequence problem, one chooses a random permutation π : [1 : n] → [1 : n] and one considers the length of its longest increasing subsequence, Ln = max{k : π[i1] < π[i2] < · · · < π[ik], where 1 ≤ i1 < i2 · · · < ik ≤ n}. On the other hand, in the sequential monotone subsequence problem one views the values π[1], π[2], . . . as though they were presented over time to a decision maker who, when shown the value π[i] at time i, must decide (once and for all) either to accept or reject π[i] as an element of the selected increasing subsequence. The decision to accept or reject π[i] at time i is based on just the knowledge of the time horizon n and the observed values π[1], π[2], . . . , π[i]. Thus, in slightly more formal language, the sequential selection problem amounts to the consideration of random variables of the form (1) Ln = max{k : π[τ1] < π[τ2] < · · · < π[τk], where 1 ≤ τ1 < τ2 · · · < τk ≤ n}, where the indices τi, i = 1, 2, . . ., are stopping times with respect to the increasing sequence of σ-fields Fk = σ{π[1], π[2], . . . , π[k] }, 1 ≤ k ≤ n. We call a sequence of such stopping times a feasible selection strategy, and, if we use τ as a shorthand for such a strategy, then the quantity of central interest here can be written as (2) s(n) = sup τ E[Ln], where one takes the supremum over all feasible selection strategies. Received by the editors September 15, 2015 and, in revised form, January 2, 2016 and January 14, 2016. 2010 Mathematics Subject Classification. Primary 60C05, 60G40, 90C40; Secondary 60F99, 90C27, 90C39.