Hall’s Theorem and SDRs

1.7 Matchings

1.7.2 Hall’s Theorem and SDRs

I’ll match that!

— Monty Hall, Let’s Make a Deal In this section we consider several classic results concerning matchings. We begin with a few more definitions.

Given a graphGand a matchingM, anM-alternating path is a path inGwhere the edges alternate betweenM-edges and non-M-edges. AnM-augmenting path is anM-alternating path where both end vertices areM-unsaturated.

As an example, consider the graphGand the matchingM indicated in Fig- ure 1.107. An example of anM-alternating path isc,a,d,e,i. An example of anM-augmenting path isj,g,f,a,c,b. The reason for calling such a path “M- augmenting” will become apparent soon.

j a

b c d

e

g f i

h

FIGURE 1.107. The graphGand matchingM. The following result is due to Berge [23].

Theorem 1.50 (Berge’s Theorem). LetM be a matching in a graphG.M is maximum if and only ifGcontains noM-augmenting paths.

Proof. First, assume that M is a maximum matching, and suppose that P : v₁, v₂, . . . , v_k is an M-augmenting path. Due to the alternating nature of M- augmenting paths, it must be thatk is even and that the edgesv₂v₃,v₄v₅,. . ., vk−2vk−1are all edges ofM. We also see that the edgesv1v2,v3v4,. . .,vk−1vk

are not edges ofM (Figure 1.108).

But then if we define the set of edgesM1to be

M1= (M \ {v2v3, . . . , vk−2vk−1})∪ {v1v2, . . . , vk−1vk},

v

1

v

2

v

3

v

4

v

5

v

6

v

k - 2

v

k - 1

v

k

FIGURE 1.108. AnM-augmenting path.

thenM1is a matching that contains one more edge thanM, a matching that we assumed to be maximum. This is a contradiction, and we can conclude thatG contains noM-augmenting paths.

For the converse, assume thatGhas noM-augmenting paths, and suppose that Mis a matching that is larger thanM. Define a subgraphHofGas follows: Let V(H) =V(G)and letE(H)be the set of edges ofGthat appear in exactly one ofM andM. Now consider some properties of this subgraphH. Since each of the vertices ofGlies on at most one edge fromMand at most one edge fromM, it must be that the degree (inH) of each vertex ofH is at most 2. This implies that each connected component ofH is either a single vertex, a path, or a cycle.

If a component is a cycle, then it must be an even cycle, since the edges alternate betweenM-edges and M-edges. So, since|M| > |M|, there must be at least one component ofH that is a path that begins and ends with edges fromM. But this path is anM-augmenting path, contradicting our assumption. Therefore, no such matchingMcan exist—implying thatM is maximum.

Before we see Hall’s classic matching theorem, we need to define one more term. IfGis a bipartite graph with partite setsX andY, we say thatX can be matched intoY if there exists a matching inGthat saturates the vertices ofX.

Consider the two examples in Figure 1.109. In the bipartite graph on the left,

X

X Y Y

FIGURE 1.109.

we see thatX can be matched into Y. In the graph on the right, though, it is impossible to matchXintoY (why is this?). What conditions on a bipartite graph must exist if we want to match one partite set into the other? The answer to this question is found in the following result of Hall [147] (Philip, not Monty).

Recall that the neighborhood of a set of vertices S, denoted byN(S), is the union of the neighborhoods of the vertices ofS.

Theorem 1.51 (Hall’s Theorem). LetGbe a bipartite graph with partite setsX andY.X can be matched intoY if and only if|N(S)| ≥ |S|for all subsetsSof X.

Proof. First suppose thatX can be matched intoY, and letSbe some subset of X. SinceSitself is also matched intoY, we see immediately that|S| ≤ |N(S)| (see Figure 1.110). Now suppose that|N(S)| ≥ |S|for all subsetsSofX, and

N(S) Y X

S

FIGURE 1.110.

letM be a maximum matching. Suppose thatu∈Xis not saturated byM (see Figure 1.111). Define the setAto be the set of vertices ofGthat can be joined tou

u

Y X

FIGURE 1.111.

by anM-alternating path. LetS=A∩X, and letT =A∩Y (see Figure 1.112).

Notice now that Berge’s Theorem implies that every vertex ofT is saturated by

Y S

X u

T

FIGURE 1.112.

M and thatuis the only unsaturated vertex ofS. That is, every vertex ofT is saturated, and every vertex ofS\{u}is saturated. This implies that|T|=|S|−1.

It follows from Berge’s Theorem and the definition ofTthatN(S) =T. But then we have that|N(S)| =|S| −1 <|S|, and this is a contradiction. We conclude that such a vertexucannot exist inXand thatMsaturates all ofX.

Given some family of setsX, a system of distinct representatives, or SDR, for the sets inXcan be thought of as a “representative” collection of distinct elements from the sets ofX. For instance, letS₁,S₂,S₃,S₄, andS₅be defined as follows:

S₁={2,8}, S2={8}, S3={5,7}, S4={2,4,8}, S5={2,4}.

The familyX1 = {S1, S2, S3, S4}does have an SDR, namely{2,8,7,4}. The familyX2={S1, S2, S4, S5}does not have an SDR.

So under what conditions will a finite family of sets have an SDR? We answer this question with the following theorem.

Theorem 1.52. LetS₁,S₂, . . . ,S_k be a collection of finite, nonempty sets. This collection has an SDR if and only if for everyt ∈ {1, . . . , k}, the union of anyt of these sets contains at leasttelements.

Proof. Since each of the sets is finite, then of courseS =S1∪S2∪ · · · ∪Sk is finite. Let us say that the elements ofSarea1, . . . ,an.

We now construct a bipartite graph with partite setsX = {S1, . . . , Sk}and Y = {a1, . . . , an} (Figure 1.113). We place an edge betweenSi andaj if and only ifaj∈Si.

Y

X S

1

S

2

S

k

a

1

a

2

a

n

FIGURE 1.113. Constructing a bipartite graph.

Hall’s Theorem now implies thatXcan be matched intoY if and only if|A| ≤

|N(A)|for all subsetsAofX. In other words, the collection of sets has an SDR if and only if for everyt∈ {1, . . . , k}, the union of anytof these sets contains at leasttelements.

Hall’s Theorem is often referred to as Hall’s Marriage Theorem. We will see more about this in Section 2.9.

Exercises

1. (From [56].) For the graphs of Figure 1.114, with matchingsM as shaded, find

(a) anM-alternating path that is notM-augmenting;

(b) anM-augmenting path if one exists; and, if so, use it to obtain a bigger matching.

FIGURE 1.114.

2. For each of the following families of sets, determine whether the condition of Theorem 1.52 is met. If so, then find an SDR. If not, then show how the condition is violated.

(a) {1,2,3},{2,3,4},{3,4,5},{4,5},{1,2,5} (b) {1,2,4},{2,4},{2,3},{1,2,3}

(c) {1,2},{2,3},{1,2,3},{2,3,4},{1,3},{3,4} (d) {1,2,5},{1,5},{1,2},{2,5}

(e) {1,2,3},{1,2,4},{1,3,4},{1,2,3,4},{2,3,4}

3. LetG be a bipartite graph. Show thatG has a matching of size at least

|E(G)|/Δ(G).

4. LetΘ ={S1, S2, . . . , Sr}be a family of distinct nonempty subsets of the set{1,2, . . . , n}. If theSiare all of the same cardinality, then prove that there exists an SDR ofΘ.

5. LetM₁andM₂be matchings in a bipartite graphGwith partite setsXand Y. IfS ⊆X is saturated byM₁andT ⊆Y is saturated byM₂, show that there exists a matching inGthat saturatesS∪T.

6. (From [139].) LetGbe a bipartite graph with partite setsXandY. LetδX

denote the minimum degree of the vertices inX, and letΔY denote the maximum degree of the vertices inY. Prove that ifδX ≥ΔY, then there exists a matching inGthat saturatesX.