Variable Decomposition for Prophet Inequalities and Optimal Ordering.

We introduce a new decomposition technique for random variables that maps a generic instance of the prophet inequalities problem to a new instance where all but a constant number of variables have a tractable structure that we refer to as $(\varepsilon, \delta)$-smallness. Using this technique, we make progress on several outstanding problems in the area: - We show that, even in the case of non-identical distributions, it is possible to achieve (arbitrarily close to) the optimal approximation ratio of $\beta \approx 0.745$ as long as we are allowed to remove a small constant number of distributions. - We show that for frequent instances of prophet inequalities (where each distribution reoccurs some number of times), it is possible to achieve the optimal approximation ratio of $\beta$ (improving over the previous best-known bound of $0.738$). - We give a new, simpler proof of Kertz's optimal approximation guarantee of $\beta \approx 0.745$ for prophet inequalities with i.i.d. distributions. The proof is primal-dual and simultaneously produces upper and lower bounds. - Using this decomposition in combination with a novel convex programming formulation, we construct the first Efficient PTAS for the Optimal Ordering problem.