Matroid Secretary for 2-Sums

This problem is a generalization of the classical secretary problem in which numbers arrive on-line in random order and the goal is to select a number as large as possible. In the matroid secretary problem, there is a matroid with ground set E and independent sets I, and a weight function assigning a weight w(i) to each element i ∈ E. We wish to design an algorithm which the matroid (E, I) is given in advance, but not the weights w(i). The ground set of the matroid is presented in random order to an online algorithm. The algorithm maintains an independent set S (in the matroid) selected out of elements who arrived so far. When an element i arrives, the algorithm learns the weight w(i) of the element. If S+i is an independent set, the algorithm may choose to add i to S. But if it does so i can never be delete from S in the future. If the algorithm does not add i, then i is lost and will not belong to the output. The goal of the algorithm is to output an independent set S of maximum weight. If the expected weight of the selected set (over a uniform random ordering of elements) is always within a c factor of the weight of the maximum weight basis for any assignment of weights to the ground set, we say the algorithm is c-competitive or a c-approximation.