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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 $d \mu$-almost all $x\in X$ and $d \nu$-almost all $y\in Y$.

My question is: if $c(x,y)\in \{0,1\}$ for all $x$ and $y$, and for each $x$ there exists $y_1,y_2$ such that $c(x,y_1)=0$ and $c(x,y_2)=1$ (and similarly for each $y$ there exist $x_1,x_2$ such that the equivalent condition holds) 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.

UPDATE: I added additional conditions about $c$, which guarantees that we can find solutions such that $0\leq\varphi\leq1$ and $-1\leq \psi \leq 0$. This is because we can shift $\varphi$ and $\psi$ such that $\sup_x \varphi(x)=1$, which guarantees that $\psi(y)\geq-1$. Furthermore, it must always be true that $\psi(y)\leq 0$ because otherwise there would exist $x,y$ such that $\varphi(x) + \psi(y) > 1$. This in turn guarantees that we can always assume that $\varphi(x)\geq 0$ as well.

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  • $\begingroup$ Two quick comments: First, I posted a wrong answer earlier where I simply messed up a detail, sorry! Second should "$\varphi(x) + \psi(y) \leq c(x, y)$ for almost all ..." be instead pointwise (since no measure on the product space is given)? $\endgroup$
    – Steve
    Mar 27, 2019 at 18:53
  • $\begingroup$ Thanks @Steve! I made a correction regarding your second comment. $\endgroup$ Mar 27, 2019 at 19:29
  • $\begingroup$ The fundamental theorem of optimal transport says that any measure on the product space that has its support in the c-superdifferential of a c-concave function is optimal for it's marginals. So you can try to build a counterexample by finding a phi which has other values than 1, 0, -1 and which is also c-concave. $\endgroup$
    – Dirk
    Mar 27, 2019 at 19:35
  • $\begingroup$ I think Steve meant that the inequality should hold everywhere (aren't the potentials continuous functions anyway?). $\endgroup$
    – Dirk
    Mar 27, 2019 at 19:37
  • $\begingroup$ @Dirk yes the theorem I'm looking at (Theorem 1.3 of Villani's "Topics in Optimal Transporation") says that you can assume WLOG that the potentials are continuous, but we're taking a supremum as opposed to a maximum. I'm updating the question with some additional findings now. $\endgroup$ Mar 27, 2019 at 20:08

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The strong Kantorovich duality, i.e. the existence of dual solutions, holds whenever $c$ is lower semicontinuous and real-valued, and the optimal coupling has finite cost, see Theorem 5.10(ii) in Villani "Optimal Transport old and new". In contrast, for the weak duality ("$\inf = \sup$") it suffices to assume $c$ is lower semicontinuous.

To prove 5.10(ii) one constructs an explicit $c$-concave function $\varphi$ that is finite-valued on the support of $\mu$, see formula (5.17). Beware of signs! Villani is looking at $c$-convex functions and you at $c$-concave functions.

To answer the "value part" of the question: the values of $\psi$ and $\varphi$ are only important on the support of $\mu$ and resp. $\nu$. Everywhere else you may choose them to be $-\infty$ (loosing $c$-concavity). Now some cases: If the optimal coupling $\pi$ has cost $1$ then for all $(x,y),(x',y') \in \Gamma := \operatorname{supp}\pi$ it holds $c(x,y')=1$ as otherwise $c(x',y)+c(x,y') < 2 = c(x,y)+c(x',y')$ violating $c$-cyclic monotonicty. Thus $\varphi \equiv 0 \equiv \psi-1$ is a dual solution to the problem. A similar case happens if the cost is $0$ on $\operatorname{supp}\mu \times \operatorname{supp}\nu$. For the last case, pick $(x_0,y_0) \in \Gamma$ with $c(x_0,y_0)=0$ and do the standard construction to get an integer-valued $c$-concave $\varphi$ with $\varphi(x_0)=0$. Observe if $(x_1,y_1),(x_2,y_2)\in \Gamma$ then

\begin{align*} \varphi(x_{1}) + \psi(x_1) & = c(x_{1},x_{1})\\ \varphi(x_{2}) + \psi(x_2) & = c(x_{2},x_{2}). \end{align*}

Subtracting this from $\varphi+\psi\le c$ applied to the couples $(x_1.y_2)$ and $(x_2,y_1)$ gives \begin{align*} \varphi(x_{2})-\varphi(x_{1}) & \le c(x_{2},y_{1})-c(x_{1},x_{1})\\ \varphi(x_{1})-\varphi(x_{2}) & \le c(x_{1},y_{2})-c(x_{2},y_{2}). \end{align*} The right hand side has values in $\{-1,0,1\}$. Since $\varphi(x_0)=0$ the function $\varphi$ has values in the same set. Since $1-(-1)=2$, it must have values either in $\{0,1\}$ or in $\{-1,0\}$. A similar argument as above then shows that $\psi$ needs have values in $\{-1,0\}$ or $\{0,1\}$.

Note if $c$ is not real-valued then there may not be any dual solutions between certain measures $\mu$ and $\nu$, see the preprint of S. Suhr and me on Lorentzian cost function, more explicitly Section 3.1.

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  • $\begingroup$ How does this answer the question (which asks if there are optimal potentials with taking only specific values )? $\endgroup$
    – Dirk
    Mar 28, 2019 at 14:55
  • $\begingroup$ Since the value question was "already" answered in the question itself, I didn't include it. Now there is a short statement. $\endgroup$ Mar 28, 2019 at 15:13
  • $\begingroup$ Could you give a reference that the standard construction shows that $\varphi$ is either identically $0$ or $0$ and somewhere $1$? I do not see how that follows from (5.17) in "Optimal transport old and new". $\endgroup$ Mar 28, 2019 at 21:31
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    $\begingroup$ I changed to a direct argument. $\endgroup$ Mar 29, 2019 at 8:17
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    $\begingroup$ Was the right hand side. Was just a typo. This is true as c has values 0 or 1 so the difference can have at most the mentioned three values. The inequality then shows that the oscillation of phi is at most 1. Knowing phi at x0 is 0 gives the claims. $\endgroup$ Apr 3, 2019 at 7:37

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