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{"Abstract":["We investigate prophet inequalities with competitive ratios approaching 1, seeking to generalize k-uniform matroids. We first show that large girth does not suffice: for all k, there exists a matroid of girth ≥ k and a prophet inequality instance on that matroid whose optimal competitive ratio is 1/2. Next, we show k-fold matroid unions do suffice: we provide a prophet inequality with competitive ratio 1-O(√{(log k)/k}) for any k-fold matroid union. Our prophet inequality follows from an online contention resolution scheme.\r\nThe key technical ingredient in our online contention resolution scheme is a novel bicriterion concentration inequality for arbitrary monotone 1-Lipschitz functions over independent items which may be of independent interest. Applied to our particular setting, our bicriterion concentration inequality yields "Chernoff-strength" concentration for a 1-Lipschitz function that is not (approximately) self-bounding."]} 

We present a constant-factor approximation algorithm for the Nash Social Welfare (NSW) maximization problem with subadditive valuations accessible via demand queries. More generally, we propose a framework for NSW optimization which assumes two subroutines which (1) solve a configuration-type LP under certain additional conditions, and (2) round the fractional solution with respect to utilitarian social welfare. In particular, a constant-factor approximation for submodular valuations with value queries can also be derived from our framework. 

We present a constant-factor approximation algorithm for the Nash Social Welfare (NSW) maximization problem with subadditive valuations accessible via demand queries. More generally, we propose a framework for NSW optimization which assumes two subroutines that (1) solve a conguration-type LP under certain additional conditions, and (2) round the fractional solution with respect to utilitarian social welfare. In particular, a constant-factor approximation for submodular valuations with value queries can also be derived from our framework.