“Hall's marriage theorem”的意思、由来-开放百科全书

Hall's marriage theorem, proved by {{harvs|txt|first=Philip|last=Hall|authorlink=Philip Hall|year=1935}}, is a theorem with two equivalent formulations:

Combinatorial formulation

Let

be a (possibly infinite) family of finite subsets of

, where the members of

are counted with multiplicity. (That is,

may contain the same set several times.)^[1]

A transversal for

is the image of an injective function

from

such that

is an element of the set

for every

in the family

. In other words,

selects one representative from each set in

in such a way that no two sets from

get the same representative. An alternative term for transversal is system of distinct representatives.

Restated in words, the marriage condition asserts that every subfamily

covers at least

different members of

Suppose that the marriage condition fails, i.e., that for some subcollection

Suppose, by way of contradiction, that a transversal

also exists.

The restriction of

to the offending subcollection

would be an injective function from

into

. This is impossible by the pigeonhole principle since

. Therefore no transversal can exist if the marriage condition fails.

Examples

A valid transversal would be (1, 4, 5). (Note this is not unique: (2, 1, 3) works equally well, for example.)

No valid transversal exists; the marriage condition is violated as is shown by the subcollection W = {A₂, A₃, A₄}. Here the number of sets in the subcollection |W| = 3 while the union of the three sets A₂ U A₃ U A₄ contain only 2 elements of X.

Application to marriage

The standard example of an application of the marriage theorem is to imagine two groups; one of n men, and one of n women. For each woman, there is a subset of the men, any one of which she would happily marry; and any man would be happy to marry a woman who wants to marry him. Consider whether it is possible to pair up (in marriage) the men and women so that every person is happy.

If we let A_i be the set of men that the i-th woman would be happy to marry, then the marriage theorem states that each woman can happily marry a man if and only if the collection of sets {A_i} meets the marriage condition.

Note that the marriage condition is that, for any subset

of the women, the number of men whom at least one of the women would be happy to marry,

, be at least as big as the number of women in that subset,

. It is obvious that this condition is necessary, as if it does not hold, there are not enough men to share among the

women. What is interesting is that it is also a sufficient condition.

Graph theoretic formulation

Let G be a finite bipartite graph with bipartite sets X and Y ( G:= (X + Y, E)). An X-saturating matching is a matching which covers every vertex in X.

For a subset W of X, let

denote the neighborhood of W in G, i.e. the set of all vertices in Y adjacent to some element of W. The marriage theorem in this formulation states that there is an X-saturating matching if and only if for every subset W of X:

In other words: every subset W of X has sufficiently many adjacent vertices in Y.

Proof of the graph theoretic version

Easy direction: we assume that some matching M saturates every vertex of X, and prove that Hall's condition is satisfied for all W ⊆ X. Let M(W) denote the set of all vertices in Y matched by M to a given W. By definition of a matching, |M(W)|=|W|. But M(W) ⊆ N_G(W), since all elements of M(W) are neighbours of W. So, |N_G(W)| ≥ |M(W)| and hence, |N_G(W)| ≥ |W|.

Hard direction: we assume that there is no X-saturating matching and prove that Hall's condition is violated for at least one W ⊆ X. Let M be a maximum matching, and u a vertex not saturated by M. Consider all alternating paths (i.e., paths in G alternately using edges outside and inside M) starting from u. Let the set of all points in Y connected to u by these alternating paths be Z, and the set of all points in X connected to u by these alternating paths (including u itself) be W. No maximal alternating path can end in a vertex in Y, lest it would be an augmenting path, so that we could augment M to a strictly larger matching by toggling the status (belongs to M or not) of all the edges of the path. Thus every vertex in Z is matched by M to a vertex in W \\ {u}. Conversely, every vertex v in W \\ {u} is matched by M to a vertex in Z (namely, the vertex preceding v on an alternating path ending at v). Thus, M provides a bijection of W \\ {u} and Z, which implies |W| = |Z| + 1. On the other hand, N_G(W) ⊆ Z: let v in Y be connected to a vertex w in W. If the edge (w,v) is in M, then v is in Z by the previous part of the proof, otherwise we can take an alternating path ending in w and extend it with v, getting an augmenting path and showing that v is in Z. Hence, |N_G(W)| ≤ |Z| = |W| − 1<|W|.

Constructive proof of the hard direction

The following algorithm either finds an X-saturating matching M, or finds a Hall violator W (a subset of X for which |N_G(W)| < |W|.^[2]

At each iteration, either W_k and Z_k grow by one vertex, or M grows by one edge (and W_k, Z_k are reset). Hence, this algorithm must end after at most |E||X| steps. It ends either with an X-saturating matching or with a Hall violator. The runtime complexity is similar to the Ford-Fulkerson method for finding a maximum cardinality matching.

Equivalence of the combinatorial formulation and the graph-theoretic formulation

Let S = (A₁, A₂, ..., A_n) where the A_i are finite sets which need not be distinct. Let the set X = {A₁, A₂, ..., A_n} (that is, the set of names of the elements of S) and the set Y be the union of all the elements in all the A_i.

We form a finite bipartite graph G:= (X + Y, E), with bipartite sets X and Y by joining any element in Y to each A_i which it is a member of. A transversal of S is an X-saturating matching (a matching which covers every vertex in X) of the bipartite graph G. Thus a problem in the combinatorial formulation can be easily translated to a problem in the graph-theoretic formulation.

Applications

The theorem has many other interesting "non-marital" applications. For example, take a standard deck of cards, and deal them out into 13 piles of 4 cards each. Then, using the marriage theorem, we can show that it is always possible to select exactly 1 card from each pile, such that the 13 selected cards contain exactly one card of each rank (Ace, 2, 3, ..., Queen, King).

More abstractly, let G be a group, and H be a finite subgroup of G. Then the marriage theorem can be used to show that there is a set T such that T is a transversal for both the set of left cosets and right cosets of H in G.

The marriage theorem is used in the usual proofs of the fact that an (r × n) Latin rectangle can always be extended to an ((r+1) × n) Latin rectangle when r < n, and so, ultimately to a Latin square.

Logical equivalences

This theorem is part of a collection of remarkably powerful theorems in combinatorics, all of which are related to each other in an informal sense in that it is more straightforward to prove one of these theorems from another of them than from first principles. These include:

In particular,^[4]^[5] there are simple proofs of the implications Dilworth's theorem ⇔ Hall's theorem ⇔ König–Egerváry theorem ⇔ König's theorem.

Infinite families

Marshall Hall Jr. variant

By examining Philip Hall's original proof carefully, Marshall Hall, Jr. (no relation to Philip Hall) was able to tweak the result in a way that permitted the proof to work for infinite S.^[6] This variant refines the marriage theorem and provides a lower bound on the number of transversals that a given S may have. This variant is:^[7]

Suppose that (A₁, A₂, ..., A_n), where the A_i are finite sets that need not be distinct, is a family of sets satisfying the marriage condition, and suppose that |A_i| ≥ r for i = 1, ..., n. Then the number of different transversals for the family is at least r ! if r ≤ n and r(r - 1) ... (r - n +1) if r > n.

Recall that a transversal for a family S is an ordered sequence, so two different transversals could have exactly the same elements. For instance, the family A₁ = {1,2,3}, A₂ = {1,2,5} has both (1,2) and (2,1) as distinct transversals.

Marriage condition does not extend

The following example, due to Marshall Hall, Jr., shows that the marriage condition will not guarantee the existence of a transversal in an infinite family in which infinite sets are allowed.

Let S be the family, A₀ = {1, 2, 3, ...}, A₁ = {1}, A₂ = {2}, ..., A_i = {i}, ...

The marriage condition holds for this infinite family, but no transversal can be constructed.^[8]

The more general problem of selecting a (not necessarily distinct) element from each of a collection of nonempty sets (without restriction as to the number of sets or the size of the sets) is permitted in general only if the axiom of choice is accepted.

The marriage theorem does extend to the infinite case if stated properly. Given a bipartite graph with sides A and B, we say that a subset C of B is smaller than or equal in size to a subset D of A in the graph if there exists an injection in the graph (namely, using only edges of the graph) from C to D, and that it is strictly smaller in the graph if in addition there is no injection in the graph in the other direction. Note that omitting in the graph yields the ordinary notion of comparing cardinalities. The infinite marriage theorem states that there exists an injection from A to B in the graph, if and only if there is no subset C of A such that N(C) is strictly smaller than C in the graph.^[9]

General graphs

A generalization of Hall's theorem to general graphs (that are not necessarily bipartite) is provided by the Tutte theorem.

Notes

1. ^{{harvnb|Hall, Jr.|1986|loc=pg. 51}}. It is also possible to have infinite sets in the family, but the number of sets in the family must then be finite, counted with multiplicity. Only the situation of having an infinite number of sets while allowing infinite sets is not allowed.
2. ^{{Cite web|url=https://www.cse.ust.hk/~golin/COMP572/Notes/Matching.pdf|title=Bipartite Matching & the Hungarian Method|last=Mordecai J. Golin|date=2006|website=|archive-url=|archive-date=|dead-url=|access-date=}}
3. ^The naming of this theorem is inconsistent in the literature. There is the result concerning matchings in bipartite graphs and its interpretation as a covering of (0,1)-matrices. {{harvtxt|Hall, Jr.|1986}} and {{harvtxt|van Lint|Wilson|1992}} refer to the matrix form as König's theorem, while {{harvtxt|Roberts|Tesman|2009}} refer to this version as the Kőnig-Egerváry theorem. The bipartite graph version is called Kőnig's theorem by {{harvtxt|Cameron|1994}} and {{harvtxt|Roberts|Tesman|2009}}.
4. ^Equivalence of seven major theorems in combinatorics
5. ^{{harvnb|Reichmeider|1984}}
6. ^{{harvnb|Hall, Jr.|1986|loc=pg. 51}}
7. ^{{harvnb|Cameron|1994|loc=pg.90}}
8. ^{{harvnb|Hall, Jr.|1986|loc=pg. 51}}
9. ^{{Cite journal|last=Aharoni|first=Ron|date=February 1984|title=König's Duality Theorem for Infinite Bipartite Graphs|journal=Journal of the London Mathematical Society|volume=s2-29|issue=1|pages=1–12|doi=10.1112/jlms/s2-29.1.1|issn=0024-6107}}

Combinatorial formulation

Examples

Application to marriage

Graph theoretic formulation

Proof of the graph theoretic version

Constructive proof of the hard direction

Equivalence of the combinatorial formulation and the graph-theoretic formulation

Applications

Logical equivalences

Infinite families

Marshall Hall Jr. variant

Marriage condition does not extend

General graphs

Notes

References

External links