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

M. Szusterman

TL;DR

The paper develops a framework linking extension complexity of polytopes to the width of Markovian randomized communication protocols. By recasting slack-matrix factorizations through k-round branching-program protocols, it proves xc$(P)=\min_{\pi}\,|\Gamma(\pi)|$ for markovian protocols, enabling concrete upper bounds via protocol design. It provides a new bound xc$(P_{match}(G))\le\tilde{O}(n^3\cdot 1.5^n)$ for the matching polytope, improving on the classic $2^n$ bound, and recovers Goemans’ compact permutahedron extension through a one-round protocol based on sorting networks, tying to the $O(n\log n)$ comparator constructions. The appendix further develops the supporting theory, including Lovász’s vertex-cover analysis, and discusses sorting-networks definitions, minimality, and Goemans extensions, highlighting open questions about exact nonnegative-rank and minimal sorting networks.

Abstract

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.

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

TL;DR

The paper develops a framework linking extension complexity of polytopes to the width of Markovian randomized communication protocols. By recasting slack-matrix factorizations through k-round branching-program protocols, it proves xc for markovian protocols, enabling concrete upper bounds via protocol design. It provides a new bound xc for the matching polytope, improving on the classic bound, and recovers Goemans’ compact permutahedron extension through a one-round protocol based on sorting networks, tying to the comparator constructions. The appendix further develops the supporting theory, including Lovász’s vertex-cover analysis, and discusses sorting-networks definitions, minimality, and Goemans extensions, highlighting open questions about exact nonnegative-rank and minimal sorting networks.

Abstract

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 for the extension complexity of the matching polytope , improving upon the standard -bound given by Edmonds' description. Additionally, we recover Goemans' compact formulation for the permutahedron using a one-round protocol based on sorting networks.
Paper Structure (13 sections, 12 theorems, 53 equations, 2 figures)

This paper contains 13 sections, 12 theorems, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Markovian protocols with branching programs
  3. A protocol for the matching polytope
  4. Correctness of the protocol given for the matching polytope
  5. Sorting networks give protocols for the permutahedron
  6. A one-round protocol for guessing slacks of the permutahedron
  7. Appendix
  8. Appendix on Markovian protocols
  9. Lovasz analysis of the greedy algorithm for vertex cover, and consequences for matchings.
  10. Deducing a non-negative factorization from the $4$-round protocol of section \ref{['section:PMG']}, page 6
  11. Equivalent definitions of sorting networks
  12. A remark about minimal sorting networks
  13. About Goemans extension of the permutahedron

Key Result

Theorem 1

Let $P\subset \mathbb{R}^d$ be a polytope. Then xc$(P)=\text{rk}_+(P).$

Figures (2)

  • Figure 1: Quoting a Markovian protocol for the spanning tree polytope
  • Figure 2: An almost Markovian protocol for the matching polytope

Theorems & Definitions (28)

  • Theorem 1: Yannakakis
  • Theorem 2: FFGT -faenzaetal
  • Theorem 3
  • Theorem 4
  • Definition 1
  • Definition 2
  • Proposition 1
  • proof : Proof of correctness
  • Claim 1
  • proof
  • ...and 18 more