  We consider applications involving a large set of instances of projecting
points to polytopes. We develop an intuition guided by theoretical and
empirical analysis to show that when these instances follow certain structures,
a large majority of the projections lie on vertices of the polytopes. To do
these projections efficiently we derive a vertex-oriented incremental algorithm
to project a point onto any arbitrary polytope, as well as give specific
algorithms to cater to simplex projection and polytopes where the unit box is
cut by planes. Such settings are especially useful in web-scale applications
such as optimal matching or allocation problems. Several such problems in
internet marketplaces (e-commerce, ride-sharing, food delivery, professional
services, advertising, etc.), can be formulated as Linear Programs (LP) with
such polytope constraints that require a projection step in the overall
optimization process. We show that in the very recent work, the polytopic
projection is the most expensive step and our efficient projection algorithms
help in gaining massive improvements in performance.
