Advances on Matroid Secretary Problems: Free Order Model and Laminar Case

The most important open conjecture in the context of the matroid secretary problem claims the existence of an O(1)-competitive algorithm applicable to any matroid. Whereas this conjecture remains open, modified forms of it have been shown to be true, when assuming that the assignment of weights to the secretaries is not adversarial but uniformly at random [23, 20]. However, so far, no variant of the matroid secretary problem with adversarial weight assignment is known that admits an O(1)-competitive algorithm. We address this point by presenting a 4-competitive procedure for the free order model, a model suggested shortly after the introduction of the matroid secretary problem, and for which no O(1)-competitive algorithm was known so far. The free order model is a relaxed version of the original matroid secretary problem, with the only difference that one can choose the order in which secretaries are interviewed. Furthermore, we consider the classical matroid secretary problem for the special case of laminar matroids. Only recently, an O(1)-competitive algorithm has been found for this case, using a clever but rather involved method and analysis [13] that leads to a competitive ratio of 16000/3. This is arguably one of the most involved special cases of the matroid secretary problem for which an O(1)-competitive algorithm is known. We present a considerably simpler and stronger 3 √ 3e ≈ 14.12-competitive procedure, based on reducing the problem to a matroid secretary problem on a partition matroid. Furthermore, our procedure is order-oblivious, which, as shown in [1], allows for transforming it into a 3 √ 3e-competitive algorithm for single-sample prophet inequalities.