In the Matroid Secretary Problem, introduced by Babaioff et al. [5] , the elements of a given matroid are presented to an online algorithm in random order. When an element is revealed, the algorithm learns its weight and decides whether or not to select it. The objective is to return a maximum weight independent set of the matroid. There are different variants for this problem depending on the information known about the weights beforehand. In the random assignment model, a hidden list of weights is randomly assigned to the matroid ground set, independently from the random order they are revealed to the algorithm. Our main result is the first constant competitive algorithm for this version of the problem, solving an open question of Babaioff et al. Our algorithm achieves a competitive ratio of 2e 2 /(e − 1). It exploits the notion of principal partition of a matroid, its decomposition into uniformly dense minors, and a 2e-competitive algorithm for uniformly dense matroids we also develop. We also present constant competitive algorithms in the standard model where the weights are assigned adversarially, for various classes of matroids including cographic, low density, k-column sparse linear matroids and the case when every element is in a small cocircuit. In the same model, we give a new O(log r)-competitive algorithm for matroids of rank r which only uses the relative order of the weights seen and not their actual values, as previously needed.
For general matroids, Soto REF developed a constant-competitive algorithm in the random assignment model, i.e., the weights of the elements are assigned uniformly at random.
Matroid secretary problem in the random assignment model
SODA '11
Mathematics Computer Science