Multi-Marginal Monge Problem

• The multi-marginal Monge problem is a deterministic formulation of optimal transport that seeks measurable maps transferring one marginal to others while minimizing total cost.
• It examines when an optimal Kantorovich plan concentrates on a graph, using conditions like twist, c-splitting, and cyclical monotonicity to ensure uniqueness.
• Special cost classes, including quadratic and symmetric costs, illustrate regimes where the optimal plan may be uniquely graph-based or decomposed into multiple graphs.

The multi-marginal Monge problem is the deterministic branch of multi-marginal optimal transport. Given marginals $\mu_1,\dots,\mu_m$ on spaces $X_1,\dots,X_m$ and a cost $c:X_1\times\cdots\times X_m\to\mathbb{R}$, it asks for measurable maps $T_i:X_1\to X_i$, $i=2,\dots,m$, such that $(T_i)_\#\mu_1=\mu_i$ and the graph-induced plan $(\mathrm{Id},T_2,\dots,T_m)_\#\mu_1$ minimizes the total cost. Its relaxed counterpart is the multi-marginal Kantorovich problem, which minimizes over all couplings $\gamma\in\Pi(\mu_1,\dots,\mu_m)$. The central structural question is when an optimal Kantorovich plan is concentrated on a graph, when it is unique, and when the best one can hope for is concentration on several graphs or on a higher-dimensional set (Pass, 2014, Kim et al., 2013).

1. Variational formulation and basic geometry

For Polish spaces or smooth manifolds $X_i$ with prescribed marginals $\mu_i$, the Kantorovich problem is

$X_1,\dots,X_m$0

while the Monge formulation restricts admissible couplings to plans of the form

$X_1,\dots,X_m$1

Thus the Monge problem is a constrained version of the Kantorovich problem: it minimizes over deterministic couplings, whereas Kantorovich allows arbitrary plans with the prescribed marginals (Griessler, 2016, Pass, 2010).

A Monge solution exists precisely when an optimal Kantorovich plan is supported on a graph over the first marginal. This is straightforward in formal terms but highly nontrivial in structure, because for $X_1,\dots,X_m$2 the geometry of $X_1,\dots,X_m$3 differs sharply from the two-marginal case. The literature repeatedly emphasizes that the general multi-marginal Monge problem is largely open for $X_1,\dots,X_m$4, even though important positive classes are known, most notably the quadratic cost of Gangbo–Świȩch (Ghoussoub et al., 2012).

The same problem is often written in maximization form when one works with a surplus $X_1,\dots,X_m$5 instead of a cost $X_1,\dots,X_m$6. In that convention one maximizes $X_1,\dots,X_m$7 over $X_1,\dots,X_m$8, and a Monge solution is again a graph-induced optimizer. This dual terminology is standard in graph-based and symmetric formulations (Pass et al., 2021).

2. Twist, splitting sets, and uniqueness of graphical optimizers

The decisive objects are $X_1,\dots,X_m$9-splitting sets. A set $c:X_1\times\cdots\times X_m\to\mathbb{R}$0 is $c:X_1\times\cdots\times X_m\to\mathbb{R}$1-splitting if there exist functions $c:X_1\times\cdots\times X_m\to\mathbb{R}$2 such that

$c:X_1\times\cdots\times X_m\to\mathbb{R}$3

everywhere, with equality on $c:X_1\times\cdots\times X_m\to\mathbb{R}$4. Optimal plans are supported on such sets, so the Monge problem is governed not only by the global geometry of $c:X_1\times\cdots\times X_m\to\mathbb{R}$5 but by the behavior of $c:X_1\times\cdots\times X_m\to\mathbb{R}$6 along splitting sets (Kim et al., 2013).

Kim–Pass introduced the condition that $c:X_1\times\cdots\times X_m\to\mathbb{R}$7 be twisted on $c:X_1\times\cdots\times X_m\to\mathbb{R}$8-splitting sets: for fixed $c:X_1\times\cdots\times X_m\to\mathbb{R}$9, the map

$T_i:X_1\to X_i$0

must be injective on each $T_i:X_1\to X_i$1-splitting set contained in $T_i:X_1\to X_i$2. Under continuity and semi-concavity of $T_i:X_1\to X_i$3, and absolute continuity of $T_i:X_1\to X_i$4, this condition implies that the optimal plan is unique and concentrated on the graph of a measurable map, hence yields a unique Monge solution (Kim et al., 2013).

Pass had earlier obtained a differential criterion of the same flavor. In his theorem, $T_i:X_1\to X_i$5, $T_i:X_1\to X_i$6-non-degeneracy, $T_i:X_1\to X_i$7-twist, and negative definiteness of a global covariant $T_i:X_1\to X_i$8-tensor $T_i:X_1\to X_i$9, together with the condition that $i=2,\dots,m$0 does not charge sets of Hausdorff dimension $i=2,\dots,m$1, imply that every Kantorovich optimizer is concentrated on a single graph. Consequently, both the Monge and Kantorovich problems have unique solutions (Pass, 2010).

Later work weakened the cost-side assumptions by strengthening the marginal regularity assumptions. Pass–Vargas-Jiménez introduced twist on $i=2,\dots,m$2-splitting sets with respect to variables $i=2,\dots,m$3: injectivity of $i=2,\dots,m$4 is required only on refined contact sets where the corresponding dual potentials are differentiable. If $i=2,\dots,m$5 are absolutely continuous, then the optimal plan is again unique and graph-concentrated. When only $i=2,\dots,m$6 is assumed absolutely continuous, this condition collapses back to the original twist on splitting sets (Pass et al., 2022).

3. Cyclical monotonicity, duality, and concentration on several graphs

In the multi-marginal setting, $i=2,\dots,m$7-cyclical monotonicity requires that for any finite family $i=2,\dots,m$8 in a set $i=2,\dots,m$9 and any permutations $(T_i)_\#\mu_1=\mu_i$0,

$(T_i)_\#\mu_1=\mu_i$1

This is the natural extension of the two-marginal notion: one fixes the first coordinate and independently reshuffles the remaining marginals (Griessler, 2016).

Griessler proved that, under the assumptions that $(T_i)_\#\mu_1=\mu_i$2 is continuous, non-negative, and bounded above by a sum of integrable functions $(T_i)_\#\mu_1=\mu_i$3, multi-marginal $(T_i)_\#\mu_1=\mu_i$4-cyclical monotonicity is sufficient for optimality. The proof passes through $(T_i)_\#\mu_1=\mu_i$5-splitting sets and dual potentials: every $(T_i)_\#\mu_1=\mu_i$6-cyclically monotone set is shown to be $(T_i)_\#\mu_1=\mu_i$7-splitting, and then the dual inequality yields optimality. In the Monge context, this turns $(T_i)_\#\mu_1=\mu_i$8-cyclical monotonicity of a candidate graph into a direct sufficient criterion for Monge optimality (Griessler, 2016).

A weaker conclusion than a single graph is often the correct one. Moameni’s measure-theoretic approach introduced $(T_i)_\#\mu_1=\mu_i$9-twist and generalized twist on $(\mathrm{Id},T_2,\dots,T_m)_\#\mu_1$0-splitting sets. Under $(\mathrm{Id},T_2,\dots,T_m)_\#\mu_1$1-twist, any optimal plan $(\mathrm{Id},T_2,\dots,T_m)_\#\mu_1$2 admits a decomposition

$(\mathrm{Id},T_2,\dots,T_m)_\#\mu_1$3

with measurable weights $(\mathrm{Id},T_2,\dots,T_m)_\#\mu_1$4 summing to $(\mathrm{Id},T_2,\dots,T_m)_\#\mu_1$5 $(\mathrm{Id},T_2,\dots,T_m)_\#\mu_1$6-a.e.; generalized twist yields an analogous countable decomposition. Thus the support is a finite or countable union of graphs over the first marginal, even when a single Monge graph does not exist (Moameni, 2014).

The several-graph picture was further developed by Moameni in a study of plans concentrated on finitely or countably many graphs. There local differential conditions implying local $(\mathrm{Id},T_2,\dots,T_m)_\#\mu_1$7-rectifiability of the support were shown to imply a local $(\mathrm{Id},T_2,\dots,T_m)_\#\mu_1$8-twist property, and hence generalized twist under compactness. The paper also analyzed extremality and uniqueness for plans supported on several graphs, showing that multi-graph concentration is a genuine intermediate regime between Monge determinism and fully non-graphical optimal transport (Moameni et al., 2015).

4. Special cost classes: quadratic, symmetric, and graph-structured models

The benchmark positive example is the Gangbo–Świȩch quadratic cost

$(\mathrm{Id},T_2,\dots,T_m)_\#\mu_1$9

Under finite second moments and the condition that each $\gamma\in\Pi(\mu_1,\dots,\mu_m)$0 vanishes on $\gamma\in\Pi(\mu_1,\dots,\mu_m)$1-rectifiable sets, Gangbo–Świȩch proved existence of a unique optimal plan, and that plan is supported on a graph: $\gamma\in\Pi(\mu_1,\dots,\mu_m)$2 The maps are expressed through convex potentials $\gamma\in\Pi(\mu_1,\dots,\mu_m)$3, and the problem is equivalent to a Wasserstein barycenter problem: the barycenter measure $\gamma\in\Pi(\mu_1,\dots,\mu_m)$4 minimizes $\gamma\in\Pi(\mu_1,\dots,\mu_m)$5, and the maps $\gamma\in\Pi(\mu_1,\dots,\mu_m)$6 are Brenier maps from $\gamma\in\Pi(\mu_1,\dots,\mu_m)$7 to $\gamma\in\Pi(\mu_1,\dots,\mu_m)$8 (Ghoussoub et al., 2012).

Symmetry introduces a different but equally rigid structure. In symmetric Monge–Kantorovich problems on $\gamma\in\Pi(\mu_1,\dots,\mu_m)$9, with equal marginals $X_i$0 and cyclic invariance, Ghoussoub–Moameni studied costs

$X_i$1

generated by vector fields. Their theorem states that the symmetric optimizer is of Monge type: $X_i$2 for a measure-preserving transformation $X_i$3 satisfying $X_i$4 a.e. The accompanying Hamiltonian representation

$X_i$5

connects symmetric OT, $X_i$6-cyclic monotonicity, and polar decomposition of vector fields (Ghoussoub et al., 2012).

Another structured family is given by graph-type surpluses

$X_i$7

where $X_i$8 is the edge set of a graph. Pass–Vargas used graph theory to classify when such surpluses admit Monge and unique solutions. Disconnected graphs, or graphs missing a crucial edge from the reference vertex, produce non-Monge and non-unique behavior. By contrast, complete graphs, graphs in which each vertex misses at most one edge, graphs with an inner hub, and certain tree-like gluings of clique-interaction subgraphs admit Monge solutions and uniqueness once $X_i$9 and selected additional marginals are absolutely continuous (Pass et al., 2021).

5. Failure of the Monge ansatz and minimal counterexamples

A common misconception is that benign costs, especially quadratic ones, should always favor Monge solutions. The discrete theory shows otherwise. Friesecke–Vögler constructed a transparent counterexample in the smallest symmetric finite setting: $\mu_i$0 marginals, $\mu_i$1 sites, and a symmetric pairwise Frenkel–Kontorova cost. The symmetric Kantorovich polytope has $\mu_i$2 extreme points, only $\mu_i$3 of which are Monge, and the unique minimizer is non-Monge. By superposition, the same mechanism yields a continuous one-dimensional example in which the Monge infimum is not attained and minimizing sequences develop microstructure (Friesecke, 2018).

For uniform discrete marginals and the quadratic multi-marginal cost

$\mu_i$4

a sharp counterexample was obtained in 2024. For $\mu_i$5, $\mu_i$6, $\mu_i$7, explicit $\mu_i$8-empirical marginals in $\mu_i$9 admit an optimal coupling with cost $X_1,\dots,X_m$00, while the best Monge-type coupling among the $X_1,\dots,X_m$01 permutation-based candidates has cost $X_1,\dots,X_m$02. Hence the Monge ansatz fails. The same paper proves that Monge always holds in three important regimes: $X_1,\dots,X_m$03 for any $X_1,\dots,X_m$04, $X_1,\dots,X_m$05 for any $X_1,\dots,X_m$06, and $X_1,\dots,X_m$07 for any $X_1,\dots,X_m$08. It follows that $X_1,\dots,X_m$09 is the smallest possible triple for failure, and that the set $X_1,\dots,X_m$10 of $X_1,\dots,X_m$11-empirical measures is not barycentrically convex for $X_1,\dots,X_m$12, $X_1,\dots,X_m$13, $X_1,\dots,X_m$14 (Emami et al., 2024).

Symmetry also obstructs uniqueness. In a symmetric multi-marginal problem with equal marginals and permutation-invariant cost, if an optimizer charges a product $X_1,\dots,X_m$15 of pairwise disjoint sets with positive mass, then permutation symmetry generates another optimizer with the same cost. In that sense, uniqueness in symmetric multi-marginal transport occurs only under very special circumstances (Moameni et al., 2015).

6. Applications, current extensions, and open structure

The multi-marginal Monge problem appears in several application areas. In economics, costs of matching-for-teams or hedonic pricing type can be written as

$X_1,\dots,X_m$16

and under twist and non-degeneracy hypotheses this yields unique Monge solutions. In the Riemannian quadratic case $X_1,\dots,X_m$17, the same formalism is equivalent to Wasserstein barycenters, so existence and uniqueness of the barycenter transport structure become a special case of multi-marginal Monge theory (Pass, 2010, Pass, 2014).

In density functional theory, the semi-classical limit produces multi-marginal OT with Coulomb cost. The survey literature emphasizes that the two-electron case admits a unique Monge solution under mild assumptions, while for $X_1,\dots,X_m$18 the picture changes drastically: in one dimension the unique symmetric minimizer is supported on the union of graphs of powers of a map, and in higher dimensions the support can have dimension at least $X_1,\dots,X_m$19. A different DFT-derived problem, based on an intermolecular interaction cost

$X_1,\dots,X_m$20

admits a unique Monge solution provided $X_1,\dots,X_m$21 and one $X_1,\dots,X_m$22-marginal are absolutely continuous; if all $X_1,\dots,X_m$23-marginals are Dirac, every coupling is optimal, showing that the extra regularity assumption is necessary for uniqueness (Pass, 2014, Gerolin et al., 2024).

Recent work has also turned the Monge problem into an explicit computational target. A deep-learning approach based on Hilbert space embeddings and MMD penalties parameterizes the maps $X_1,\dots,X_m$24 by neural networks and solves a penalized multi-marginal Monge problem directly on GPUs; for the pairwise quadratic cost, the paper proves that the MMD penalties enforce the marginal constraints asymptotically (Nakano et al., 12 Jul 2025). A complementary dynamical formulation rewrites multi-marginal OT with convex or semi-convex translation-invariant costs as a convex optimization problem on flows of couplings over $X_1,\dots,X_m$25, with a primal-dual structure and numerically computed quasi-Monge solutions (Pass et al., 26 Sep 2025).

Across these developments, a stable dichotomy emerges. For special costs—quadratic barycentric costs, twist-compatible one-dimensional costs, several graph-structured surpluses, and some DFT-derived anisotropic bilinear costs—optimal plans are deterministic or finitely multi-valued. Outside those classes, especially under symmetry or discreteness, optimal transport may concentrate on several graphs, on high-dimensional sets, or on nonattained limits with microstructure. The multi-marginal Monge problem is therefore less a single theorem than a hierarchy of structural regimes, organized by twist, splitting geometry, cyclical monotonicity, marginal regularity, and symmetry.