The submodular secretary problem under a cardinality constraint and with limited resources.

We study the submodular secretary problem subject to a cardinality constraint, in long-running scenarios, or under resource constraints. In these scenarios the resources consumed by the algorithm should not grow with the input size, and the online selection algorithm should be an anytime algorithm. We propose a $0.1933$-competitive anytime algorithm, which performs only a single evaluation of the marginal contribution for each observed item, and requires a memory of order only $k$ (up to logarithmic factors), where $k$ is the cardinality constraint. The best competitive ratio for this problem known so far under these constraints is $frac{e-1}{e^2+e} approx 0.1700$ (Feldman et al., 2011). Our algorithm is based on the observation that information collected during times in which no good items were selected, can be used to improve the subsequent probability of selection success. The improvement is obtained by using an adaptive selection strategy, which is a solution to a stand-alone online selection problem. We develop general tools for analyzing this algorithmic framework, which we believe will be useful also for other online selection problems.