Markovian protocols and an upper bound on the extension complexity of the matching polytope

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.

💡 Research Summary

The paper studies the extension complexity (xc) of polytopes through a novel lens that connects non‑negative factorizations of slack matrices with a special class of randomized communication protocols called Markovian protocols. The classical result of Yannakakis equates xc(P) with the non‑negative rank of the slack matrix of P, and Faenza‑Fiorini‑Grapp‑Tiwary later related this rank to the cost of a binary‑tree communication protocol (the height of the tree). However, tree‑based representations obscure the role of the number of communication rounds and can be suboptimal for certain polytopes.

The authors introduce a branching‑program (BP) model for communication. A Markovian protocol proceeds in k rounds; each round j has a finite set of states V_j and a transition probability matrix p^{(j)} that depends only on the current state and the player whose turn it is. Because the protocol is Markovian, the future evolution depends solely on the present state, not on the full transcript. For a given protocol π, they define the width |Γ(π)| as the number of state‑tuples (u_1,…,u_k) that can occur with positive probability for some inputs. This width measures the total number of distinct paths through the BP.

The central theoretical contribution (Theorem 3) shows that \