We study two-sided many-to-one matching markets with transferable utilities in which money can exchange hands between matched agents, subject to distributional constraints on the set of feasible allocations. In such markets, we establish that equilibrium arrangements are surplus-maximizing and study the conditions on the distributional constraints under which equilibria exist and can be computed efficiently when agents have linear preferences. Our main result is a linear programming duality method to efficiently compute equilibrium arrangements under sufficient conditions on the constraint structure guaranteeing equilibrium existence. This linear programming approach provides a method to compute market equilibria in polynomial time in the number of firms, workers, and the cardinality of the constraint set.