An Improved Lower Bound for Matroid Intersection Prophet Inequalities
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We consider prophet inequalities subject to feasibility constraints that are the intersection of q matroids. The best-known algorithms achieve a Θ(q)-approximation, even when restricted to instances that are the intersection of q partition matroids, and with i.i.d. Bernoulli random variables. The previous best-known lower bound is Θ(√(q)) due to a simple construction of [Kleinberg-Weinberg STOC 2012] (which uses i.i.d. Bernoulli random variables, and writes the construction as the intersection of partition matroids). We establish an improved lower bound of q^1/2+Ω(1/loglog q) by writing the construction of [Kleinberg-Weinberg STOC 2012] as the intersection of asymptotically fewer partition matroids. We accomplish this via an improved upper bound on the product dimension of a graph with p^p disjoint cliques of size p, using recent techniques developed in [Alon-Alweiss European Journal of Combinatorics 2020].
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