Improved competitive ratio for the matroid secretary problem

The Matroid Secretary Problem, introduced by Babaioff et al. (2007), is a generalization of the Classical Secretary Problem. In this problem, elements from a matroid are presented to an on-line algorithm in a random order. Each element has a weight associated with it, which is revealed to the algorithm along with the element. After each element is revealed the algorithm must make an irrevocable decision on whether or not to select it. The goal is to pick an independent set with the sum of the weights of the selected elements as large as possible. Babaioff et al gave an algorithm for the Matroid Secretary Problem with a competitive ratio of O(logρ), where ρ is the rank of the matroid. It has been conjectured that a constant competitive-ratio is achievable for this problem. In this paper we give an algorithm that has a competitive-ratio of O(√logρ).