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 [José R. Correa et al., 2022; Moran Feldman et al., 2016; Marek Adamczyk and Michal Wlodarczyk, 2018]. The previous best-known lower bound is Θ(√q) due to a simple construction of [Robert Kleinberg and S. Matthew Weinberg, 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/log log q)} by writing the construction of [Robert Kleinberg and S. Matthew Weinberg, 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 [Noga Alon and Ryan Alweiss, 2020].
@InProceedings{saxena_et_al:LIPIcs.ITCS.2023.95, author = {Saxena, Raghuvansh R. and Velusamy, Santhoshini and Weinberg, S. Matthew}, title = {{An Improved Lower Bound for Matroid Intersection Prophet Inequalities}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {95:1--95:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.95}, URN = {urn:nbn:de:0030-drops-175986}, doi = {10.4230/LIPIcs.ITCS.2023.95}, annote = {Keywords: Prophet Inequalities, Intersection of Matroids} }
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