Consider the usual Monge-Kantorovich transportation problem 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 Kantorovich duality theorem states that the transportation cost between $\mu$ and $\nu$ is equal to the supremum of $$\int_X \varphi~ d\mu +\int_Y \psi~ d\nu $$ over all $L_1$ functions $\varphi(x)$ and $\psi(y)$ such that $\varphi(x)+\psi(y)\leq c(x,y)$ for almost all $x\in X$ and $y\in Y$.

My question is: if $c(x,y)\in \{0,1\}$ for all $x$ and $y$, does it follow that there exists a solution (or "almost exists" a solution) where $\varphi(x)$ and $\psi(x)$ only take values in the set $\{-1,0,1\}$? Finite dimensional experiments with linear programs suggest that the answer is "yes" but I cannot tell if they extend to the general setting.