  We prove a strong duality result for a linear programming problem which has
the interpretation of being a discretised optimal Skorokhod embedding problem,
and we recover this continuous time problem as a limit of the discrete
problems. With the discrete setup we show that for a suitably chosen objective
function, the optimiser takes the form of a hitting time for a random walk. In
the limiting problem we then reprove the existence of the Root, Rost, and cave
embedding solutions of the Skorokhod embedding problem.
  The main strength of this approach is that we can derive properties of the
discrete problem more easily than in continuous time, and then prove that these
properties hold in the limit. For example, the strong duality result gives dual
optimisers, and our limiting arguments can be used to derive properties of the
continuous time dual functions, known to represent a superhedging portfolio.
