Submodular Secretary Problems: Cardinality, Matching, and Linear Constraints

We study various generalizations of the secretary problem with submodular objective functions. Generally, a set of requests is revealed step-by-step to an algorithm in random order. For each request, one option has to be selected so as to maximize a monotone submodular function while ensuring feasibility. For our results, we assume that we are given an offline algorithm computing an α-approximation for the respective problem. This way, we separate computational limitations from the ones due to the online nature. When only focusing on the online aspect, we can assume α = 1. In the submodular secretary problem, feasibility constraints are cardinality constraints. That is, out of a randomly ordered stream of entities, one has to select a subset size k. For this problem, we present a 0.31α-competitive algorithm for all k, which asymptotically reaches competitive ratio fracαe for large k. In submodular secretary matching, one side of a bipartite graph is revealed online. Upon arrival, each node has to be matched permanently to an offline node or discarded irrevocably. We give an fracα4-competitive algorithm. In both cases, we improve over previously best known competitive ratios, using a generalization of the algorithm for the classic secretary problem. Furthermore, we give an O(α d^-frac2B-1)-competitive algorithm for submodular function maximization subject to linear packing constraints. Here, d is the column sparsity, that is the maximal number of none-zero entries in a column of the constraint matrix, and B is the minimal capacity of the constraints. Notably, this bound is independent of the total number of constraints. We improve the algorithm to be O(α d^-frac1B-1)-competitive if both d and B are known to the algorithm beforehand.