Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 456 — #464
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Chapter 12
Simple Graphs
Inductive Step: Suppose that jM j D m C 1 2. To find a matching for M , there
are two cases.
Case 1: Every nonempty subset of at most m men likes a strictly larger set of
women. In this case, we have some latitude: we pair an arbitrary man with
a woman he likes and send them both away. This leaves m men and one
fewer women, and the matching condition will still hold. So the induction
hypothesis P.m/ implies we can match the remaining m men.
Case 2: Some nonempty subset X of at most m men likes an equal-size set Y of
women. The matching condition must hold within X, so the strong induction
hypothesis implies we can match the men in X with the women in Y . This
leaves the problem of matching the set M X of men to the set W Y of
women.
But the problem of matching M X against W Y also satisfies the Matching
condition, because any subset of men in M X who liked fewer women
in W Y would imply there was a set of men who liked fewer women in the
whole set W . Namely, if a subset M 0 M X liked only a strictly smaller
subset of women W 0 W Y , then the set M 0 [ X of men would like only
women in the strictly smaller set W 0 [ Y . So again the strong induction hypothesis
implies we can match the men in M X with the women in W Y ,
which completes a matching for M .
So in both cases, there is a matching for the men, which completes the proof of
the Inductive step. The theorem follows by induction.
The proof of Theorem 12.5.2 gives an algorithm for finding a matching in a bipartite
graph, albeit not a very efficient one. However, efficient algorithms for finding
a matching in a bipartite graph do exist. Thus, if a problem can be reduced to
finding a matching, instances of the problem can be solved in a reasonably efficient
way.
A Formal Statement
Let’s restate Theorem 12.5.2 in abstract terms so that you’ll not always be condemned
to saying, “Now this group of men likes at least as many women. . . ”
Definition 12.5.3. A matching in a graph G is a set M of edges of G such that no
vertex is an endpoint of more than one edge in M . A matching is said to cover a
set S of vertices iff each vertex in S is an endpoint of an edge of the matching. A
matching is said to be perfect if it covers V .G/. In any graph G the set N.S/ of