[Paper Review] Monge solutions and uniqueness in multi-marginal optimal transport via graph theory

Brendan Pass, Adolfo Vargas-Jiménez|arXiv (Cornell University)|Apr 19, 2021

American Literature and Culture4 citations

TL;DR

This paper establishes uniqueness and Monge solution results for multi-marginal optimal transport problems with a class of surplus functions defined via graph structures. By associating the surplus with a graph on m vertices, the authors prove that when the graph satisfies certain connectivity and degree conditions—such as each vertex having degree m−1 or m−2—solutions are unique and induced by deterministic maps, generalizing known results for complete and cycle graphs.

ABSTRACT

We study a multi-marginal optimal transport problem with surplus $b(x_{1}, \ldots, x_{m})=\sum_{\{i,j\}\in P} x_{i}\cdot x_{j}$, where $P\subseteq Q:=\{\{i,j\}: i, j \in \{1,2,...m\}, i eq j\}$. We reformulate this problem by associating each surplus of this type with a graph with $m$ vertices whose set of edges is indexed by $P$. We then establish uniqueness and Monge solution results for two general classes of surplus functions. Among many other examples, these classes encapsulate the Gangbo and Świȩch surplus [12] and the surplus $\sum_{i=1}^{m-1}x_{i}\cdot x_{i+1} + x_{m}\cdot x_{1}$ studied in an earlier work by the present authors [23].

Motivation & Objective

• To address the challenge of determining when multi-marginal optimal transport problems admit unique Monge solutions, especially when standard twist conditions fail.
• To generalize existing uniqueness and Monge solution results beyond the twist-on-splitting-sets condition, which is hard to verify in practice.
• To unify and extend known results for specific surplus functions—such as the Gangbo–Świȩch and cyclic surpluses—by introducing a graph-theoretic framework.
• To characterize the structural conditions on the surplus (encoded as a graph) that guarantee Monge solutions and uniqueness under absolute continuity of marginals.

Proposed method

• Reformulate the multi-marginal optimal transport problem by associating the surplus function $ b(x_1, \ldots, x_m) = \sum_{\{i,j\} \in P} x_i \cdot x_j $ with a graph $ G $ having $ m $ vertices and edge set $ P $.
• Define key graph-theoretic properties: vertex degree $ |N(v)| \in \{m-1, m-2\} $, maximal cliques $ S_j $, and inner hubs $ A $, to classify the structure of the surplus.
• Use a novel application of Lemma 2.1 to compare two solutions $ x^1 $ and $ x^2 $, proving equality on $ I(V(G) \setminus V(S)) $, $ I(V(S_k)) $, and ultimately $ x^1 = x^2 $ everywhere.
• Apply the method of contradiction and case analysis on vertex neighborhoods and cliques to show that equalities in marginal sums imply pointwise equality of transport plans.
• Leverage absolute continuity of $ \mu_1 $ (or $ \mu_i $ for some $ i \in I(N(v_1)) $) to ensure the existence of a unique deterministic map solution.
• Generalize known results: the Gangbo–Świȩch surplus (complete graph) and the cyclic surplus (cycle graph) are special cases of the framework.

Experimental results

Research questions

• RQ1 Under what graph-theoretic conditions on the surplus function $ b(x_1, \ldots, x_m) = \sum_{\{i,j\} \in P} x_i \cdot x_j $ does the multi-marginal Kantorovich problem admit a unique Monge solution?
• RQ2 Can the classical twist-on-splitting-sets condition be relaxed or generalized using graph structure, particularly for non-twisted surpluses?
• RQ3 How do structural properties of the graph—such as vertex degree and clique decomposition—determine the existence and uniqueness of Monge solutions?
• RQ4 Can the results for the Gangbo–Świȩch and cyclic surpluses be unified and extended under a single graph-theoretic framework?
• RQ5 What role does absolute continuity of the first marginal $ \mu_1 $, or of some $ \mu_i $, play in ensuring Monge solutions when the twist condition fails?

Key findings

• For any surplus function associated with a graph $ G $ on $ m $ vertices where each vertex has degree $ m-1 $ or $ m-2 $, and with $ \mu_1 $ absolutely continuous, every solution to the Kantorovich problem is induced by a deterministic map.
• The result generalizes the Gangbo–Świȩch surplus (complete graph) and the cyclic surplus $ \sum_{i=1}^{m-1} x_i \cdot x_{i+1} + x_m \cdot x_1 $, both of which are special cases of the framework.
• When $ m \leq 4 $, the cyclic surplus admits unique Monge solutions even when the twist-on-splitting-sets condition fails, provided $ \mu_1 $ is absolutely continuous.
• For $ m \geq 5 $, the cyclic surplus may admit non-Monge solutions, but the graph-theoretic condition still ensures uniqueness and Monge form under absolute continuity.
• The proof technique relies on comparing two solutions via sum constraints over vertex neighborhoods and using a refined version of Lemma 2.1 to force pointwise equality across the graph.
• The framework allows for graphs with missing edges (e.g., $ C_m \setminus \text{edges} $) as long as each vertex is missing at most one neighbor, preserving the structural conditions for uniqueness.