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Scalable real-time assortment optimization has become essential in e-commerce operations due to the need for personalization and the availability of a large variety of items. While this can be done when there are simplistic assortment choices to be made, the optimization process becomes difficult when imposing constraints on the collection of relevant assortments based on insights by store-managers and historically well-performing assortments. We design fast and flexible algorithms based on variations of binary search that find the (approximately) optimal assortment in this difficult regime. In particular, we revisit the problem of large-scale assortment optimization under the multinomial logit choice model without any assumptions on the structure of the feasible assortments. We speed up the comparison steps using advances in similarity search in the field of information retrieval/machine learning. For an arbitrary collection of assortments, our algorithms can find a solution in time that is sub-linear in the number of assortments, and for the simpler case of cardinality constraints - linear in the number of items (existing methods are quadratic or worse). Empirical validations using a real world dataset (in addition to experiments using semi-synthetic data based on the Billion Prices dataset and several retail transaction datasets) show that our algorithms are competitive even when the number of items is $\sim 10^5$ ($10\times$ larger instances than previously studied).