$c$-cyclical monotonicity as a sufficient criterion for optimality in   the multi-marginal Monge-Kantorovich problem
  This note establishes that a generalization of $c$-cyclical monotonicity from the Monge-Kantorovich problem with two marginals gives rise to a sufficient condition for optimality also in the multi-marginal version of that problem. To obtain the result, the cost function is assumed to be bounded by a sum of integrable functions. The proof rests on ideas from martingale transport. 
Claus Griessler
math.OC