Online hypergraph matching: hiring teams of secretaries

The goal of this paper is to find a competitive algorithm for the following problem. We are given a hypergraph G = (X,E), |E| = n, with weighted edges and maximum edge size k. We wish to maximize the weight of a matching M ⊆ E of G, given that edges are revealed in a random order, and that when an edge e is revealed the algorithm must decide either e ∈ M or e / ∈M . Note that this is a generalization of the classic secretary problem; given a set {wi} of weights for the candidates in the secretary problem, we create a hypergraph with one vertex v and n edges, where the ith edge ei = {v} has cardinality 1 and weight wi. Clearly a matching can be only one edge ei, corresponding to choosing candidate i. In fact, when k = 1, the hypergraph matching problem is just a collection of independent secretary problems, one for each vertex.