In this article, we propose a generalized weighted version of the well-known\nBenjamini-Hochberg (BH) procedure. The rigorous weighting scheme used by our\nmethod enables it to encode structural information from simultaneous multi-way\nclassification as well as hierarchical partitioning of hypotheses into groups,\nwith provisions to accommodate overlapping groups. The method is proven to\ncontrol the False Discovery Rate (FDR) when the p-values involved are\nPositively Regression Dependent on the Subset (PRDS) of null p-values. A\ndata-adaptive version of the method is proposed. Simulations show that our\nproposed methods control FDR at desired level and are more powerful than\nexisting comparable multiple testing procedures, when the p-values are\nindependent or satisfy certain dependence conditions. We apply this\ndata-adaptive method to analyze a neuro-imaging dataset and understand the\nimpact of alcoholism on human brain. Neuro-imaging data typically have complex\nclassification structure, which have not been fully utilized in subsequent\ninference by previously proposed multiple testing procedures. With a flexible\nweighting scheme, our method is poised to extract more information from the\ndata and use it to perform a more informed and efficient test of the\nhypotheses.\n