Consider the usual transport problem of Monge-Kantorovich where $ X $ and $ Y $ are Polish spaces, $ mu $ and $ nu $ are probability measures on $ X $ and $ Y $, and $ c: X times Y to mathbb {R} ^ + cup {+ infty } $ is a lower semi-continuous cost function. The theorem of Kantorovich's duality states that the cost of transportation between $ mu $ and $ nu $ is equal to the supremum of $$ int_X varphi ~ d mu + int_Y psi ~ d nu $$ mostly $ L_1 $ the functions $ varphi (x) $ and $ psi (y) $ such as $ varphi (x) + psi (y) leq c (x, y) $ for almost everyone $ x in X $ and $ y in Y $.

My question is: if $ c (x, y) in {0,1 } $ for everyone $ x $ and $ y $Does this result in a solution (or solution "almost exists") where $ varphi (x) $ and $ psi (x) $ only take values as a whole $ {- 1,0,1 } $? Finite dimensional experiments with linear programs suggest that the answer is "yes" but I can not say whether they extend to the general framework.