Matching polytope

In graph theory, the matching polytope of a given graph is a geometric object representing the possible matchings in the graph. It is a convex polytope each of whose corners corresponds to a matching. It has great theoretical importance in the theory of matching.^{[1]: 273–285}

Let G = (V, E) be a graph with n = |V| nodes and m = |E| edges.

For every subset U of vertices, its incidence vector 1_U is a vector of size n, in which element v is 1 if node v is in U, and 0 otherwise. Similarly, for every subset F of edges, its incidence vector 1_F is a vector of size m, in which element e is 1 if edge e is in F, and 0 otherwise.

For every node v in V, the set of edges in E adjacent to v is denoted by E(v). Therefore, each vector 1_E(v) is a 1-by-m vector in which element e is 1 if edge e is adjacent to v, and 0 otherwise. The incidence matrix of the graph, denoted by A_G, is an n-by-m matrix in which each row v is the incidence vector 1_E(V). In other words, each element v,e in the matrix is 1 if node v is adjacent to edge e, and 0 otherwise.

Below are three examples of incidence matrices: the triangle graph (a cycle of length 3), a square graph (a cycle of length 4), and the complete graph on 4 vertices.

${\displaystyle {\begin{pmatrix}1&1&0\\1&0&1\\0&1&1\end{pmatrix}}~,~{\begin{pmatrix}1&1&0&0\\0&1&1&0\\0&0&1&1\\1&0&0&1\\\end{pmatrix}}~,~{\begin{pmatrix}1&1&0&0&1&0\\0&1&1&0&0&1\\0&0&1&1&1&0\\1&0&0&1&0&1\\\end{pmatrix}}}$

For every subset F of edges, the dot product 1_E(v) · 1_F represents the number of edges in F that are adjacent to v. Therefore, the following statements are equivalent:

• A subset F of edges represents a matching in G;
• For every node v in V: 1_E(v) · 1_F ≤ 1.
• A_G · 1_F ≤ 1_V.

The cardinality of a set F of edges is the dot product 1_E · 1_{F .} Therefore, a maximum cardinality matching in G is given by the following integer linear program:

Maximize 1_E · x

Subject to: x in {0,1}^m

__________ A_G · x ≤ 1_V.

The fractional matching polytope of a graph G, denoted FMP(G), is the polytope defined by the relaxation of the above linear program, in which each x may be a fraction and not just an integer:

Maximize 1_E · x

Subject to: x ≥ 0_E

__________ A_G · x ≤ 1_V.

This is a linear program. It has m "at-least-0" constraints and n "less-than-one" constraints. The set of its feasible solutions is a convex polytope. Each point in this polytope is a fractional matching. For example, in the triangle graph there are 3 edges, and the corresponding linear program has the following 6 constraints:

Maximize x₁+x₂+x₃

Subject to: x₁≥0, x₂≥0, x₃≥0_.

__________ x₁+x₂≤1, x₂+x₃≤1, x₃+x₁≤1.

This set of inequalities represents a polytope in R³ - the 3-dimensional Euclidean space.

The polytope has five corners (extreme points). These are the points that attain equality in 3 out of the 6 defining inequalities. The corners are (0,0,0), (1,0,0), (0,1,0), (0,0,1), and (1/2,1/2,1/2).^[2] The first corner (0,0,0) represents the trivial (empty) matching. The next three corners (1,0,0), (0,1,0), (0,0,1) represent the three matchings of size 1. The fifth corner (1/2,1/2,1/2) does not represent a matching - it represents a fractional matching in which each edge is "half in, half out". Note that this is the largest fractional matching in this graph - its weight is 3/2, in contrast to the three integral matchings whose size is only 1.

As another example, in the 4-cycle there are 4 edges. The corresponding LP has 4+4=8 constraints. The FMP is a convex polytope in R⁴. The corners of this polytope are (0,0,0,0), (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1), (1,0,1,0), (0,1,0,1). Each of the last 2 corners represents matching of size 2, which is a maximum matching. Note that in this case all corners have integer coordinates.

The integral matching polytope (usually called just the matching polytope) of a graph G, denoted MP(G), is a polytope whose corners are the incidence vectors of the integral matchings in G.

MP(G) is always contained in FMP(G). In the above examples:

• The MP of the triangle graph is strictly contained in its FMP, since the MP does not contain the non-integral corner (1/2, 1/2, 1/2).
• The MP of the 4-cycle graph is identical to its FMP, since all the corners of the FMP are integral.

The above example is a special case of the following general theorem:^[1]: 274

G is a bipartite graph if-and-only-if MP(G) = FMP(G) if-and-only-if all corners of FMP(G) have only integer coordinates.

This theorem can be proved in several ways.

When G is bipartite, its incidence matrix A_G is totally unimodular - every square submatrix of it has determinant 0, +1 or −1. The proof is by induction on k - the size of the submatrix (which we denote by K). The base k = 1 follows from the definition of A_G - every element in it is either 0 or 1. For k>1 there are several cases:

• If K has a column consisting only of zeros, then det K = 0.
• If K has a column with a single 1, then det K can be expanded about this column and it equals either +1 or -1 times a determinant of a (k − 1) by (k − 1) matrix, which by the induction assumption is 0 or +1 or −1.
• Otherwise, each column in K has two 1s. Since the graph is bipartite, the rows can be partitioned into two subsets, such that in each column, one 1 is in the top subset and the other 1 is in the bottom subset. This means that the sum of the top subset and the sum of the bottom subset are both equal to 1_E minus a vector of |E| ones. This means that the rows of K are linearly dependent, so det K = 0.

As an example, in the 4-cycle (which is bipartite), the det A_G = 1. In contrast, in the 3-cycle (which is not bipartite), det A_G = 2.

Each corner of FMP(G) satisfies a set of m linearly-independent inequalities with equality. Therefore, to calculate the corner coordinates we have to solve a system of equations defined by a square submatrix of A_G. By Cramer's rule, the solution is a rational number in which the denominator is the determinant of this submatrix. This determinant must by +1 or −1; therefore the solution is an integer vector. Therefore all corner coordinates are integers.

By the n "less-than-one" constraints, the corner coordinates are either 0 or 1; therefore each corner is the incidence vector of an integral matching in G. Hence FMP(G) = MP(G).

A facet of a polytope is the set of its points which satisfy an essential defining inequality of the polytope with equality. If the polytope is d-dimensional, then its facets are (d − 1)-dimensional. For any graph G, the facets of MP(G) are given by the following inequalities:^{[1]: 275–279}

• x ≥ 0_E
• 1_E(v) · x ≤ 1 (where v is a non-isolated vertex such that, if v has only one neighbor u, then {u,v} is a connected component of G, and if v has exactly two neighbors, then they are not adjacent).
• 1_E(S) · x ≤ (|S| − 1)/2 (where S spans a 2-connected factor-critical subgraph.)

The perfect matching polytope of a graph G, denoted PMP(G), is a polytope whose corners are the incidence vectors of the integral perfect matchings in G.^[1]: 274 Obviously, PMP(G) is contained in MP(G); In fact, PMP(G) is the face of MP(G) determined by the equality:

1_E · x = n/2.

Edmonds^[3] proved that, for every graph G, PMP(G) can be described by the following constraints:

1_E(v) · x = 1 for all v in V (-- exactly one edge adjacent to v is in the matching)

1_E(W) · x ≥ 1 for every subset W of V with |W| odd (-- at least one edge should connect W to V\W). These constraints are called odd cut constraints.

x ≥ 0_E

Using this characterization and Farkas lemma, it is possible to obtain a good characterization of graphs having a perfect matching.^[4]: 206 By solving algorithmic problems on convex sets, one can find a minimum-weight perfect matching.^{[4]: 206--208}

• Stable matching polytope

• The matching polytope, by Michael Goemans
• The matching polytope, by Jan Vondrak
• The matching polytope, by Vincent Jost