Perfect Matching Polytope Overview

• The perfect matching polytope is the convex hull of incidence vectors representing perfect matchings in a graph, defined by degree and odd-cut constraints.
• Its facet structure, governed by exponentially many odd-set inequalities, reveals deep links to combinatorial optimization and graph decomposition techniques.
• Despite the underlying matching problem being in P, the polytope’s exponential extension complexity influences approximation methods and algorithm design.

The perfect matching polytope is the convex hull of incidence vectors of perfect matchings of a graph. For a graph $G=(V,E)$ with an even number of vertices, and in particular for a matching-covered graph, it is defined by

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$

where $\chi^M\in\{0,1\}^E$ is the incidence vector of $M$ (Abdi et al., 21 Aug 2025). In the complete-graph setting, this polytope is one of the central objects of combinatorial optimization: its vertices encode perfect matchings, its facet structure is governed by odd cuts or, equivalently in related formulations, odd-set constraints, and its geometry connects polyhedral theory, matching-covered graph structure, extension complexity, NC algorithms, and lattice questions (Rothvoss, 2013).

1. Definitions and classical descriptions

For the complete graph on $n$ vertices, with $n$ even, the perfect matching polytope is

$$P_{PM}(n) = \operatorname{conv}\{\chi_M \in \mathbb{R}^E \mid M \subseteq E \text{ is a perfect matching}\}.$$

Edmonds’ description becomes

$$P_{PM}(n) = \left\{ x \in \mathbb{R}^E \,\middle|\, x(\delta(v)) = 1 \ \forall v\in V;\; x(\delta(U)) \ge 1 \ \forall U\subseteq V,\ |U|\text{ odd};\; x_e \ge 0\ \forall e\in E \right\},$$

where $\delta(U)$ is the set of edges with exactly one endpoint in $U$ (Rothvoss, 2013). For a matching-covered graph $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$0, Edmonds and Johnson describe $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$1 by nonnegativity constraints $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$2, degree equations $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$3, and odd-cut inequalities $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$4 for every odd set $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$5 (Abdi et al., 21 Aug 2025).

The odd-cut formulation is dual in spirit to the odd-set formulation for the matching polytope. For the complete graph $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$6, the matching polytope is

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$7

with linear description

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$8

The odd set constraints

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$9

are the key exponentially many facets (Sinha, 2017). The perfect matching polytope is the exact-degree analogue of this description.

In the bipartite case, the description simplifies sharply. For a bipartite graph $\chi^M\in\{0,1\}^E$0, the bipartite perfect matching polytope

$\chi^M\in\{0,1\}^E$1

has the compact description

$\chi^M\in\{0,1\}^E$2

and this description is totally unimodular (Wulf, 23 Feb 2025). This distinction between the general and bipartite cases is fundamental throughout the subject.

2. Facets, odd cuts, and structural graph theory

The facet structure of matching and perfect matching polytopes is controlled by graph structure. For a graph $\chi^M\in\{0,1\}^E$3, Edmonds and Pulleyblank give a complete facet description of the matching polytope: $\chi^M\in\{0,1\}^E$4 and Lovász’s theorem states

$\chi^M\in\{0,1\}^E$5

(Benchetrit et al., 2015). Thus nontrivial facet inequalities are governed by odd ear-decompositions of induced subgraphs.

The parameter $\chi^M\in\{0,1\}^E$6 measures this complexity. For a 2-connected graph $\chi^M\in\{0,1\}^E$7, $\chi^M\in\{0,1\}^E$8 is the largest integer $\chi^M\in\{0,1\}^E$9 such that $M$0 contains a subgraph $M$1 with an ear-decomposition whose first $M$2 ears are odd; equivalently, it is the maximum length of a starting sequence of odd ears in an ear-decomposition of a subgraph of $M$3 (Benchetrit et al., 2015). When $M$4, every relevant factor-critical induced subgraph is simple enough that the matching polytope is completely described by non-negativity, star inequalities, and odd-circuit inequalities (Benchetrit et al., 2015). Since

$M$5

this connects matching-polytope structure to h-perfection of line graphs (Benchetrit et al., 2015).

For perfect matchings in matching-covered graphs, tight cuts play an analogous structural role. A tight cut $M$6 is an odd cut such that

$M$7

and cut-contractions along tight cuts preserve the matching-covered structure (Abdi et al., 21 Aug 2025). Repeated contraction yields the decomposition into bricks and braces. The polytope dimension satisfies

$M$8

where $M$9 is the number of bricks in a tight-cut decomposition (Abdi et al., 21 Aug 2025).

This decomposition has a wider scope. For matching covered uniformable hypergraphs, the perfect matching polytope

$n$0

admits an analogous tight-cut theory, and if $n$1 is tight then

$n$2

(Beckenbach et al., 2018). This suggests that polyhedral decomposition along tight cuts is not merely graph-specific, although uniqueness fails outside the uniformable class (Beckenbach et al., 2018).

3. Extension complexity and approximation

A central development in the modern theory is that the perfect matching polytope has exponential extension complexity. For a polytope $n$3, the extension complexity is

$n$4

and Yannakakis’s theorem gives

$n$5

where $n$6 is a slack matrix of $n$7 (Rothvoss, 2013). For all even $n$8,

$n$9

so the perfect matching polytope in the complete $n$0-node graph does not admit a polynomial-size extended formulation (Rothvoss, 2013).

The relevant slack matrix for this lower bound has rows indexed by odd cuts $n$1, columns indexed by perfect matchings $n$2, and entries

$n$3

A standard rectangle covering argument is insufficient; the proof instead uses a hyperplane separation lower bound with a carefully chosen weight matrix $n$4 (Rothvoss, 2013). The same result yields

$n$5

through a known reduction from a face of the TSP polytope (Rothvoss, 2013).

Approximation does not make the polyhedral problem easy. For a monotone polytope $n$6, a polytope $n$7 is a $n$8-approximation if

$n$9

equivalently, for every nonnegative objective $$P_{PM}(n) = \operatorname{conv}\{\chi_M \in \mathbb{R}^E \mid M \subseteq E \text{ is a perfect matching}\}.$$0,

$$P_{PM}(n) = \operatorname{conv}\{\chi_M \in \mathbb{R}^E \mid M \subseteq E \text{ is a perfect matching}\}.$$1

(Sinha, 2017). For the matching polytope, keeping only odd-set inequalities with

$$P_{PM}(n) = \operatorname{conv}\{\chi_M \in \mathbb{R}^E \mid M \subseteq E \text{ is a perfect matching}\}.$$2

gives a $$P_{PM}(n) = \operatorname{conv}\{\chi_M \in \mathbb{R}^E \mid M \subseteq E \text{ is a perfect matching}\}.$$3-approximation with about

$$P_{PM}(n) = \operatorname{conv}\{\chi_M \in \mathbb{R}^E \mid M \subseteq E \text{ is a perfect matching}\}.$$4

facets (Sinha, 2017).

The lower bound matches this truncation up to constants in $$P_{PM}(n) = \operatorname{conv}\{\chi_M \in \mathbb{R}^E \mid M \subseteq E \text{ is a perfect matching}\}.$$5. For every

$$P_{PM}(n) = \operatorname{conv}\{\chi_M \in \mathbb{R}^E \mid M \subseteq E \text{ is a perfect matching}\}.$$6

any extended formulation $$P_{PM}(n) = \operatorname{conv}\{\chi_M \in \mathbb{R}^E \mid M \subseteq E \text{ is a perfect matching}\}.$$7 satisfying

$$P_{PM}(n) = \operatorname{conv}\{\chi_M \in \mathbb{R}^E \mid M \subseteq E \text{ is a perfect matching}\}.$$8

must have at least

$$P_{PM}(n) = \operatorname{conv}\{\chi_M \in \mathbb{R}^E \mid M \subseteq E \text{ is a perfect matching}\}.$$9

defining inequalities, where $$P_{PM}(n) = \left\{ x \in \mathbb{R}^E \,\middle|\, x(\delta(v)) = 1 \ \forall v\in V;\; x(\delta(U)) \ge 1 \ \forall U\subseteq V,\ |U|\text{ odd};\; x_e \ge 0\ \forall e\in E \right\},$$0 is an absolute constant (Sinha, 2017). The proof uses a lower bound on the non-negative rank of a lopsided unique disjointness matrix

$$P_{PM}(n) = \left\{ x \in \mathbb{R}^E \,\middle|\, x(\delta(v)) = 1 \ \forall v\in V;\; x(\delta(U)) \ge 1 \ \forall U\subseteq V,\ |U|\text{ odd};\; x_e \ge 0\ \forall e\in E \right\},$$1

showing

$$P_{PM}(n) = \left\{ x \in \mathbb{R}^E \,\middle|\, x(\delta(v)) = 1 \ \forall v\in V;\; x(\delta(U)) \ge 1 \ \forall U\subseteq V,\ |U|\text{ odd};\; x_e \ge 0\ \forall e\in E \right\},$$2

for a suitable range of $$P_{PM}(n) = \left\{ x \in \mathbb{R}^E \,\middle|\, x(\delta(v)) = 1 \ \forall v\in V;\; x(\delta(U)) \ge 1 \ \forall U\subseteq V,\ |U|\text{ odd};\; x_e \ge 0\ \forall e\in E \right\},$$3 (Sinha, 2017). The paper states explicitly that these results apply directly to the perfect matching polytope because the relevant slack matrix is built from perfect matchings (Sinha, 2017).

A recurring misconception is that polynomial-time solvability should imply compact LP formulations. The perfect matching problem is in $$P_{PM}(n) = \left\{ x \in \mathbb{R}^E \,\middle|\, x(\delta(v)) = 1 \ \forall v\in V;\; x(\delta(U)) \ge 1 \ \forall U\subseteq V,\ |U|\text{ odd};\; x_e \ge 0\ \forall e\in E \right\},$$4, yet the classical polytope has exponential extension complexity (Rothvoss, 2013). A different result constructs, for each $$P_{PM}(n) = \left\{ x \in \mathbb{R}^E \,\middle|\, x(\delta(v)) = 1 \ \forall v\in V;\; x(\delta(U)) \ge 1 \ \forall U\subseteq V,\ |U|\text{ odd};\; x_e \ge 0\ \forall e\in E \right\},$$5, a different polytope

$$P_{PM}(n) = \left\{ x \in \mathbb{R}^E \,\middle|\, x(\delta(v)) = 1 \ \forall v\in V;\; x(\delta(U)) \ge 1 \ \forall U\subseteq V,\ |U|\text{ odd};\; x_e \ge 0\ \forall e\in E \right\},$$6

where $$P_{PM}(n) = \left\{ x \in \mathbb{R}^E \,\middle|\, x(\delta(v)) = 1 \ \forall v\in V;\; x(\delta(U)) \ge 1 \ \forall U\subseteq V,\ |U|\text{ odd};\; x_e \ge 0\ \forall e\in E \right\},$$7 if the graph $$P_{PM}(n) = \left\{ x \in \mathbb{R}^E \,\middle|\, x(\delta(v)) = 1 \ \forall v\in V;\; x(\delta(U)) \ge 1 \ \forall U\subseteq V,\ |U|\text{ odd};\; x_e \ge 0\ \forall e\in E \right\},$$8 has a perfect matching and $$P_{PM}(n) = \left\{ x \in \mathbb{R}^E \,\middle|\, x(\delta(v)) = 1 \ \forall v\in V;\; x(\delta(U)) \ge 1 \ \forall U\subseteq V,\ |U|\text{ odd};\; x_e \ge 0\ \forall e\in E \right\},$$9 otherwise, and shows that $\delta(U)$0 has a polynomial-size weak extended formulation (Avis et al., 2014). This does not contradict the lower bound for Edmonds’ polytope, because the formulation is weaker than a standard extended formulation and is decision-oriented rather than a projection onto the classical perfect matching polytope (Avis et al., 2014).

4. Lattice structure and matching-covered graphs

For a matching-covered graph $\delta(U)$1, the perfect matching polytope is closely tied to the perfect matching lattice

$\delta(U)$2

the set of all integer linear combinations of incidence vectors of perfect matchings (Abdi et al., 21 Aug 2025). The comparison object is

$\delta(U)$3

Three structural facts are established polyhedrally: $\delta(U)$4 admits a lattice basis consisting only of incidence vectors of perfect matchings; if

$\delta(U)$5

then

$\delta(U)$6

and if $\delta(U)$7 has no Petersen brick in its tight-cut decomposition, then

$\delta(U)$8

(Abdi et al., 21 Aug 2025).

The obstruction in the general case is parity contributed by Petersen bricks. Writing

$\delta(U)$9

the stronger formula is

$U$0

where $U$1 is the edge set of a $U$2-cycle in the $U$3-th Petersen brick (Abdi et al., 21 Aug 2025). Thus each Petersen brick contributes one parity restriction.

A later polyhedral construction makes this basis theorem algorithmic. There is a polynomial time algorithm that, given a matching covered graph $U$4, finds a lattice basis for $U$5 consisting of incidence vectors of perfect matchings of $U$6 (Silina, 5 Nov 2025). Its strategy is to reduce via tight cut decomposition, handle Birkhoff von Neumann graphs $U$7 satisfying

$U$8

and in non-BvN bricks use separating cuts and a perfect matching $U$9 intersecting such a cut in three edges (Silina, 5 Nov 2025). The dimension formulas

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$00

make explicit how brick structure controls both the polytope and the lattice (Silina, 5 Nov 2025).

This body of work indicates that the perfect matching polytope is not only a feasible-region object for optimization; it also governs the integral linear-algebraic structure of perfect matchings in matching-covered graphs.

5. Bipartite perfect matching polytopes: geometry and hardness beyond optimization

The bipartite perfect matching polytope is often viewed as polyhedrally simple because its standard formulation is compact and totally unimodular (Wulf, 23 Feb 2025). That simplicity is exact for feasibility and linear optimization, but it does not extend to all geometric questions.

For a bipartite graph $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$01, the diameter $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$02 is the maximum shortest-path distance in the $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$03-skeleton, and the circuit diameter $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$04 replaces edge steps by circuit moves (Wulf, 23 Feb 2025). In this setting, vertices correspond to perfect matchings, and adjacency in the $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$05-skeleton satisfies Chvátal’s criterion: two perfect matchings are adjacent iff their symmetric difference is a single alternating cycle (Wulf, 23 Feb 2025). Nevertheless,

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$06

and because

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$07

the same hardness transfers to circuit diameter (Wulf, 23 Feb 2025). The paper also states that there exists $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$08 such that the diameter of the bipartite perfect matching polytope cannot be approximated better than $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$09 in polynomial time unless $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$10, with explicit value $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$11 noted in the proof (Wulf, 23 Feb 2025).

Shortest paths in the skeleton are hard even in a more local sense. Unless $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$12, there is no polynomial-time algorithm that computes a path of constant length between two vertices at distance two of the perfect matching polytope of a bipartite graph (Cardinal et al., 2022). Under ETH, there is no polynomial-time algorithm computing a path of length at most

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$13

between two vertices at distance two, even when the graph has maximum degree three (Cardinal et al., 2022). The same paper translates this into hardness for shortest monotone paths and for circuit-augmentation algorithms on $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$14 (Cardinal et al., 2022).

The perfect matching polytope is also central in parallel algorithms. For planar graphs, an NC algorithm for finding a perfect matching works by operating inside

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$15

using the standard constraints

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$16

and, crucially, computing a point

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$17

in the relative interior of a minimum-weight face (Anari et al., 2017). Tight odd sets with

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$18

then guide contractions in parallel (Anari et al., 2017). This shows that the polytope is an algorithmic device, not merely a static description.

6. Variants, generalizations, and limitations of classical intuition

Several closely related polytopes illustrate both the robustness and the limits of the classical theory.

A broad generalization is the fractional perfect $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$19-matching polytope

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$20

When $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$21 for all $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$22, this is the polytope of fractional perfect matchings (Behrend, 2013). Its facial structure is controlled by support graphs: if $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$23 is a nonempty face, then

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$24

and

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$25

where $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$26 is the number of bipartite components of $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$27 (Behrend, 2013). An element $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$28 is a vertex if and only if each connected component of $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$29 is either acyclic or contains exactly one odd cycle (Behrend, 2013). This places the perfect matching polytope inside a larger graph-structured family.

A different extension adds one binary variable $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$30 indicating whether two fixed disjoint edges $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$31 are simultaneously chosen in a bipartite matching. For the corresponding exact polytope $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$32, the complete irredundant description consists of nonnegativity, degree inequalities, $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$33, $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$34, $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$35, and two blossom-type families

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$36

and

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$37

(Walter, 2016). The perfect-matching version is obtained by replacing degree inequalities with equalities (Walter, 2016).

Parity-constrained versions are substantially harder. For complete bipartite graphs with parallel red and blue edges, the convex hull of perfect matchings with an odd number of red edges,

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$38

has exponential extension complexity: $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$39 (Jia et al., 2022). This is not a statement about the classical bipartite perfect matching polytope itself, which remains compactly described (Jia et al., 2022). The distinction is essential.

More recently, the odd-red bipartite perfect matching polytope

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$40

was shown to have complex facet structure (Nägele et al., 18 Mar 2026). The proposed label-based relaxation

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$41

is not exact and is hard to separate over: deciding whether $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$42 is $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$43-complete even when $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$44 is promised to lie in the perfect matching polytope of $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$45 (Nägele et al., 18 Mar 2026). Moreover, for every even $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$46, there exists an exponential family of facets such that every integral facet description

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$47

must satisfy

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$48

(Nägele et al., 18 Mar 2026). This rules out exact descriptions using only coefficients in $$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$49.

Finally, cardinality constraints reveal another boundary. It is well known that the intersection of the matching polytope with a cardinality constraint is integral, but a generic “perfect matching polytope + subset cardinality constraint” statement is false (Damci-Kurt et al., 2014). An explicit bipartite example yields a fractional extreme point for the perfect matching polytope intersected with

$$P(G)=\operatorname{conv}\{\,\chi^M : M \text{ is a perfect matching of } G\,\}\subseteq \mathbb{R}^E,$$50

(Damci-Kurt et al., 2014). Thus classical integrality phenomena for matchings do not transfer wholesale to perfect matchings under arbitrary side constraints.

The perfect matching polytope therefore occupies a distinctive position in polyhedral combinatorics. Its exact description is classical, but its odd-cut structure remains decisive in approximation, decomposition, and algorithm design; its extension complexity is exponential despite polynomial-time solvability; its lattice theory is governed by tight cuts, bricks, and Petersen obstructions; and even in the bipartite case, seemingly simple geometric questions such as diameter, shortest paths, and parity-restricted variants exhibit markedly harder behavior than the base polytope itself (Rothvoss, 2013).