Monotone k-submodular secretary problems: Cardinality and knapsack constraints

In this paper, we consider the k-submodular secretary problem, in which the items or secretaries arrive one by one in a uniformly random order, and the goal is to select k disjoint sets of items, so as to maximize the expectation of a monotone non-negative k-submodular function. A decision for each item must be made immediately and irrevocably after its arrival: accept and assign it to one of the k dimensions, or reject it. We first show that, in the unconstrained setting, there is an offline algorithm that can be transformed to a k2k−1-competitive online algorithm, which is asymptotically the best possible. For the problem with cardinality constraints, we present online algorithms with provable performance guarantees for the total size constraint and individual size constraint settings that depend on the corresponding budget parameters. For the problem under a knapsack constraint, we provide a constant-competitive online algorithm.