The multi-marginal Monge Kantorovich problem (MKP) has become a pivotal framework in probability theory, mathematical physics, and economics. Despite its significance, the complexity of its solution structure and uniqueness continues to pose difficulties. In this work, we examine the properties of optimal transport plans under the \(m\) -twist condition, which ensures that any optimal plan is confined to the union of graphs of at most \(m\) transport maps, offering a precise characterization of solutions. Beyond this structural result, we explore alternative formulations of (MKP) problem, including its decomposition into two-marginal transport problems, which offer a decompositional approach. Specifically, we show that (MKP) can be rewritten as the center-marginal Monge Kantorovich problem (C-MKP) or adjacent-marginal Monge Kantorovich problem (A-MKP), each of which retains structural properties from the classical two-marginal setting. Our results refine the understanding of optimality and uniqueness in (MKP), bridging between two-marginal classical optimal transport theory and multi-marginal formulations. They have implications for applications in density functional theory, economics, and quantum transport by providing rigorous conditions for when solutions are concentrated on finitely many maps. By introducing new structural conditions techniques, this work advances our understanding of multi-marginal optimal transport.
