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A Polyhedral Perspective on the Perfect Matching Lattice

Olha Silina

TL;DR

This work addresses constructing a lattice basis for the perfect matching lattice $\mathcal{L}(G)$ of a matching-covered graph $G$ using polyhedral methods. It develops a framework around the perfect matching polytope $PM(G)$ and its bipartite relaxation $P(G)$, leveraging tight-cut decompositions into bricks and braces to reduce to Birkhoff–von Neumann graphs. The authors present a polynomial-time algorithm that either yields a lattice basis consisting of perfect matchings or produces a separating odd cut to iteratively decompose the graph; Petersen obstructions are handled via a careful case analysis and a composition mechanism across cuts. By connecting polyhedral geometry with classical ear-decomposition ideas, the paper provides new tools for tackling longstanding questions in matching theory and highlights robust methods for bridging combinatorial and geometric perspectives.

Abstract

We study the perfect matching lattice of a matching covered graph $G$, generated by the incidence vectors of its perfect matchings. Building on results of Lovász and de Carvalho, Lucchesi, and Murty, we give a polynomial-time algorithm based on polyhedral methods that constructs a lattice basis for this lattice consisting of perfect matchings of $G$. By decomposing along certain odd cuts, we reduce the graph into subgraphs whose perfect matching polytopes coincide with their bipartite relaxations (known as \emph{Birkhoff von Neumann graphs}). This yields a constructive polyhedral proof of the existence of such bases and highlights new connections between combinatorial and geometric properties of perfect matchings.

A Polyhedral Perspective on the Perfect Matching Lattice

TL;DR

This work addresses constructing a lattice basis for the perfect matching lattice of a matching-covered graph using polyhedral methods. It develops a framework around the perfect matching polytope and its bipartite relaxation , leveraging tight-cut decompositions into bricks and braces to reduce to Birkhoff–von Neumann graphs. The authors present a polynomial-time algorithm that either yields a lattice basis consisting of perfect matchings or produces a separating odd cut to iteratively decompose the graph; Petersen obstructions are handled via a careful case analysis and a composition mechanism across cuts. By connecting polyhedral geometry with classical ear-decomposition ideas, the paper provides new tools for tackling longstanding questions in matching theory and highlights robust methods for bridging combinatorial and geometric perspectives.

Abstract

We study the perfect matching lattice of a matching covered graph , generated by the incidence vectors of its perfect matchings. Building on results of Lovász and de Carvalho, Lucchesi, and Murty, we give a polynomial-time algorithm based on polyhedral methods that constructs a lattice basis for this lattice consisting of perfect matchings of . By decomposing along certain odd cuts, we reduce the graph into subgraphs whose perfect matching polytopes coincide with their bipartite relaxations (known as \emph{Birkhoff von Neumann graphs}). This yields a constructive polyhedral proof of the existence of such bases and highlights new connections between combinatorial and geometric properties of perfect matchings.

Paper Structure

This paper contains 20 sections, 17 theorems, 26 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tight cut decomposition
  4. Lattice theory and dimension results
  5. Petersen graph
  6. Tight cut combining
  7. A polyhedral approach: Birkhoff von Neumann graphs
  8. An algorithm for BvN graphs
  9. Non-BvN detection
  10. General graphs
  11. Proof of Theorem \ref{['thm:find-robust']}
  12. Finding a separating cut
  13. Finding a facet defining cut
  14. Avoiding Petersen bricks after separating cuts
  15. Acknowledgments
  16. ...and 5 more sections

Key Result

Theorem 1.1

There is a polynomial time algorithm that, given a matching covered graph $G$, finds a lattice basis for $\mathcal{L}(G)$ consisting of incidence vectors of perfect matchings of $G$.

Figures (1)

  • Figure 1: The Petersen graph. Its perfect matchings are the central one (yellow) and all possible rotations of the one in blue.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 2.2: Theorem $4.7$ in edmonds1982brick
  • Theorem 2.3: Theorem $3.2$ in LovaszOntheStr
  • Corollary 2.4
  • Theorem 2.5: Theorem $6.3$ in lovaszPM
  • Definition 2.6
  • Definition 2.7: composition
  • Theorem 2.8
  • Theorem 2.9
  • Theorem 3.1
  • ...and 22 more