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We consider the problem to transport resources/mass while abiding byconstraints on the flow through constrictions along their path betweenspecified terminal distributions. Constrictions, conceptualized as tollstations at specified points, limit the flow rate across. We quantify flow-rateconstraints via a bound on a sought probability density of the times thatmass-elements cross toll stations and cast the transportation scheduling in aKantorovich-type of formalism. Recent work by our team focused on the existenceof Monge maps for similarly constrained transport minimizing average kineticenergy. The present formulation in this paper, besides being substantially moregeneral, is cast as a (generalized) multi-marginal transport problem - aproblem of considerable interest in modern-day machine learning literature andmotivated extensive computational analyses. An enabling feature of ourformalism is the representation of an average quadratic cost on the speed oftransport as a convex constraint that involves crossing times.

Monge-Kantorovich Optimal Transport Through Constrictions and Flow-rate Constraints