This paper investigates the extension complexity of polytopes by exploiting the correspondence between non-negative factorizations of slack matrices and randomized communication protocols. We introduce a geometric characterization of extension complexity based on the width of Markovian protocols, as a variant of the framework introduced by Faenza et al. This enables us to derive a new upper bound of $\tilde{O}(n^3\cdot 1.5^n)$ for the extension complexity of the matching polytope $P_{\text{match}}(n)$, improving upon the standard $2^n$-bound given by Edmonds' description. Additionally, we recover Goemans' compact formulation for the permutahedron using a one-round protocol based on sorting networks.

翻译：本文通过利用松弛矩阵的非负分解与随机化通信协议之间的对应关系，研究多面体的扩展复杂度。我们基于马尔可夫协议的宽度，引入了扩展复杂度的几何刻画，作为Faenza等人所提出框架的一个变体。这使我们能够为匹配多面体$P_{\text{match}}(n)$的扩展复杂度推导出一个新的上界$\tilde{O}(n^3\cdot 1.5^n)$，改进了由Edmonds描述给出的标准$2^n$上界。此外，我们利用基于排序网络的一轮协议，恢复了Goemans关于置换多面体的紧致表述。