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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 (ε, δ)-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 β ~0.745 when the items arrive in a random order (this version is commonly known as prophet secretary) as long as we are allowed to remove a small constant number of distributions. We show that forfrequent instances (where each distribution reoccurs some number of times) and random arrival order, it is possible to achieve the optimal approximation ratio of β (improving over the previous best-known bound of 0.738). We give a new, simpler proof of Kertz's optimal approximation guarantee of β ~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 (in parallel work with[1]) an Efficient PTAS (EPTAS) for the Optimal Ordering problem.