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Concept
Maximum cardinality matching
Summary
Maximum cardinality matching is a fundamental problem in graph theory.
We are given a graph G, and the goal is to find a matching containing as many edges as possible; that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this problem is equivalent to the task of finding a matching that covers as many vertices as possible.
An important special case of the maximum cardinality matching problem is when G is a bipartite graph, whose vertices V are partitioned between left vertices in X and right vertices in Y, and edges in E always connect a left vertex to a right vertex. In this case, the problem can be efficiently solved with simpler algorithms than in the general case.
The simplest way to compute a maximum cardinality matching is to follow the Ford–Fulkerson algorithm. This algorithm solves the more general problem of computing the maximum flow. A bipartite graph (X + Y, E) can be converted to a flow network as follows.
Add a source vertex s; add an edge from s to each vertex in X.
Add a sink vertex t; add an edge from each vertex in Y to t.
Assign a capacity of 1 to each edge.
Since each edge in the network has integral capacity, there exists a maximum flow where all flows are integers; these integers must be either 0 or 1 since the all capacities are 1. Each integral flow defines a matching in which an edge is in the matching if and only if its flow is 1. It is a matching because:
The incoming flow into each vertex in X is at most 1, so the outgoing flow is at most 1 too, so at most one edge adjacent to each vertex in X is present.
The outgoing flow from each vertex in Y is at most 1, so the incoming flow is at most 1 too, so at most one edge adjacent to each vertex in Y is present.
The Ford–Fulkerson algorithm proceeds by repeatedly finding an augmenting path from some x ∈ X to some y ∈ Y and updating the matching M by taking the symmetric difference of that path with M (assuming such a path exists).
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