Given a graph G
), a matching M
is a set of pairwise non-adjacent
edges, none of which are loops
; that is, no two edges share a common vertex.
A vertex is matched (or saturated) if it is an endpoint of one of the edges in the matching. Otherwise the vertex is unmatched.
A maximal matching is a matching M of a graph G that is not a subset of any other matching. A matching M of a graph G is maximal if every edge in G has a non-empty intersection with at least one edge in M. The following figure shows examples of maximal matchings (red) in three graphs.
A maximum matching
(also known as maximum-cardinality matching
) is a matching that contains the largest possible number of edges. There may be many maximum matchings. The matching number
of a graph
is the size of a maximum matching. Every maximum matching is maximal, but not every maximal matching is a maximum matching. The following figure shows examples of maximum matchings in the same three graphs.
A perfect matching
is a matching that matches all vertices of the graph. That is, a matching is perfect if every vertex of the graph is incident
to an edge of the matching. Every perfect matching is maximum and hence maximal. In some literature, the term complete matching
is used. In the above figure, only part (b) shows a perfect matching. A perfect matching is also a minimum-size edge cover
. Thus, the size of a maximum matching is no larger than the size of a minimum edge cover: ν(G) ≤ ρ(G)
. A graph can only contain a perfect matching when the graph has an even number of vertices.
A near-perfect matching
is one in which exactly one vertex is unmatched. Clearly, a graph can only contain a near-perfect matching when the graph has an odd number
of vertices, and near-perfect matchings are maximum matchings. In the above figure, part (c) shows a near-perfect matching. If every vertex is unmatched by some near-perfect matching, then the graph is called factor-critical
Given a matching M
, an alternating path
is a path that begins with an unmatched vertex
and whose edges belong alternately to the matching and not to the matching. An augmenting path
is an alternating path that starts from and ends on free (unmatched) vertices. Berge's lemma
states that a matching M
is maximum if and only if there is no augmenting path with respect to M
In any graph without isolated vertices, the sum of the matching number and the edge covering number
equals the number of vertices.
If there is a perfect matching, then both the matching number and the edge cover number are |V
| / 2.
If A and B are two maximal matchings, then |A| ≤ 2|B| and |B| ≤ 2|A|. To see this, observe that each edge in B \ A can be adjacent to at most two edges in A \ B because A is a matching; moreover each edge in A \ B is adjacent to an edge in B \ A by maximality of B, hence
Further we deduce that
In particular, this shows that any maximal matching is a 2-approximation of a maximum matching and also a 2-approximation of a minimum maximal matching. This inequality is tight: for example, if G is a path with 3 edges and 4 vertices, the size of a minimum maximal matching is 1 and the size of a maximum matching is 2.
A spectral characterization of the matching number of a graph is given by Hassani Monfared and Mallik as follows: Let
be a graph
on vertices, and
be distinct nonzero purely imaginary numbers
. Then the matching number
is if and only if (a) there is a real skew-symmetric matrix
zeros, and (b) all real skew-symmetric matrices with graph
have at most
nonzero eigenvalues 
. Note that the (simple) graph of a real symmetric or skew-symmetric matrix
of order has vertices and edges given by the nonozero off-diagonal entries of
A generating function
of the number of k
-edge matchings in a graph is called a matching polynomial. Let G
be a graph and mk
be the number of k
-edge matchings. One matching polynomial of G
Another definition gives the matching polynomial as
where n is the number of vertices in the graph. Each type has its uses; for more information see the article on matching polynomials.
Algorithms and computational complexity
A fundamental problem in combinatorial optimization
is finding a maximum matching
. This problem has various algorithms for different classes of graphs.
In a weighted bipartite graph,
the optimization problem is to find a maximum-weight matching; a dual problem is to find a minimum-weight matching. This problem is often called maximum weighted bipartite matching
, or the assignment problem
. The Hungarian algorithm
solves the assignment problem and it was one of the beginnings of combinatorial optimization algorithms. It uses a modified shortest path
search in the augmenting path algorithm. If the Bellman–Ford algorithm
is used for this step, the running time of the Hungarian algorithm becomes
, or the edge cost can be shifted with a potential to achieve
running time with the Dijkstra algorithm
and Fibonacci heap
A maximal matching can be found with a simple greedy algorithm
. A maximum matching is also a maximal matching, and hence it is possible to find a largest
maximal matching in polynomial time. However, no polynomial-time algorithm is known for finding a minimum maximal matching
, that is, a maximal matching that contains the smallest
possible number of edges.
A maximal matching with k
edges is an edge dominating set
edges. Conversely, if we are given a minimum edge dominating set with k
edges, we can construct a maximal matching with k
edges in polynomial time. Therefore, the problem of finding a minimum maximal matching is essentially equal to the problem of finding a minimum edge dominating set.
Both of these two optimization problems are known to be NP-hard
; the decision versions of these problems are classical examples of NP-complete
Both problems can be approximated
within factor 2 in polynomial time: simply find an arbitrary maximal matching M
The number of matchings in a graph is known as the Hosoya index
of the graph. It is #P-complete
to compute this quantity, even for bipartite graphs.
It is also #P-complete to count perfect matchings
, even in bipartite graphs
, because computing the permanent
of an arbitrary 0–1 matrix (another #P-complete problem) is the same as computing the number of perfect matchings in the bipartite graph having the given matrix as its biadjacency matrix
. However, there exists a fully polynomial time randomized approximation scheme for counting the number of bipartite matchings.
A remarkable theorem of Kasteleyn
states that the number of perfect matchings in a planar graph
can be computed exactly in polynomial time via the FKT algorithm
Finding all maximally-matchable edges
One of the basic problems in matching theory is to find in a given graph all edges that may be extended to a maximum matching in the graph (such edges are called maximally-matchable edges
, or allowed
edges). Algorithms for this problem include:
- For general graphs, a deterministic algorithm in time and a randomized algorithm in time .
- For bipartite graphs, if a single maximum matching is found, a deterministic algorithm runs in time .
Online bipartite matching
In the online setting, nodes on one side of the bipartite graph arrive one at a time and must either be immediately matched to the other side of the graph or discarded. This is a natural generalization of the secretary problem
and has applications to online ad auctions. The best online algorithm, for the unweighted maximization case with a random arrival model, attains a competitive ratio
Matching in general graphs
Matching in bipartite graphs
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Last edited on 7 May 2021, at 22:23
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