Matching Totally Explained

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ť This article is about mathematical matchings. For other senses of this word, see matching (disambiguation).

In the mathematical discipline of graph theory a matching or edge independent set in a graph is a set of edges without common vertices. It may also be an entire graph consisting of edges without common vertices.

Definition

Given a graph G = (V,E), a matching M in G is a set of pairwise non-adjacent edges; that is, no two edges share a common vertex.
   We say that a vertex is matched if it's incident to an edge in the matching. Otherwise the vertex is unmatched.
   A maximal matching is a matching M of a graph G with the property that if any edge not in M is added to M, it's no longer a matching, that is, M is maximal if it isn't a proper subset of any other matching in graph G. In other words, 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.
   A maximum 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. Note that every maximum matching must be maximal, but not every maximal matching must be maximum.
   A perfect matching is a matching which covers all vertices of the graph. That is, every vertex of the graph is incident to exactly one edge of the matching. Every perfect matching is maximum and hence maximal. In some literature, the term complete matching is used.
   Given a matching M,

an alternating path is a path in which the edges belong alternatively to the matching and not to the matching.
an augmenting path is an alternating path that starts from and ends on free (unmatched) vertices.

One can prove that a matching is maximum if and only if it doesn't have any augmenting path.

Matchings in bipartite graphs

Matching problems are often concerned with bipartite graphs. Finding a maximum bipartite matching (often called a maximum cardinality bipartite matching) in a bipartite graph

G=(V=(X,Y),E)

is perhaps the simplest problem. The augmenting path algorithm finds it by finding an augmenting path from each

x in X

Y

and adding it to the matching if it exists. As each path can be found in

O(E)

time, the running time is

O(V E)

. This solution is equivalent to adding a super source

s

with edges to all vertices in

X

, and a super sink

t

with edges from all vertices in

Y

, and finding a maximal flow from

s

t

. All edges with flow from

X

Y

then constitute a maximum matching. An improvement over this is the Hopcroft-Karp algorithm, which runs in

O(sqrt E)

time.
In a weighted bipartite graph, each edge has an associated value. A maximum weighted bipartite matching is defined as a perfect matching where the sum of the values of the edges in the matching have a maximal value. If the graph isn't complete bipartite, missing edges are inserted with value zero. Finding such a matching is known as the assignment problem. You can solve it by using a modified shortest path search in the augmenting path algorithm. If you use the Bellman-Ford algorithm, the running time becomes

O(V^2 E)

, or you can shift the edge cost with a potential and achieve

O(V^2 log(V) + V E)

running time with the Dijkstra algorithm and Fibonacci heap. The remarkable Hungarian algorithm solves the assignment problem and it was one of the starting point of the combinatorial optimization. The original approach of this algorithm need

O(V^2E)

running time, but it could be improved to

O(V^2 log(V) + V E)

time with extensive use of priority queues. König's theorem states that, in bipartite graphs, the maximum matching is equal in size to the minimum vertex cover. Via this result, the minimum vertex cover problem and maximum independent set problem may be solved in polynomial time for bipartite graphs.

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