On Variants of the Matroid Secretary Problem

Abstract

We present a number of positive and negative results for variants of the matroid secretary problem. Most notably, we design a constant-factor competitive algorithm for the random assignment model where the weights are assigned randomly to the elements of a matroid, and then the elements arrive on-line in an adversarial order (extending a result of Soto [20]). This is under the assumption that the matroid is known in advance. If the matroid is unknown in advance, we present an O(log r log n)-approximation, and prove that a better than O(log n/ log log n) approximation is impossible. This resolves an open question posed by Babaioff et al. [3]. As a natural special case, we also consider the classical secretary problem where the number of candidates n is unknown in advance. If n is chosen by an adversary from {1,...,N}, we provide a nearly tight answer, by providing an algorithm that chooses the best candidate with probability at least 1/(HN-1 + 1) and prove that a probability better than 1/HN cannot be achieved (where HN is the N-th harmonic number).