Transversals of kernels

In this section, our main result is we prove that there exists a transversal of F G-kernels that will will be crucial in Chapter 5. For convenience, let KF G denote the set of F G-kernels for choice functions F,G : 2^E →2^E.

Theorem 3.7 (Fleiner [20]). For any modular and strictly monotone function w : E → R⁺ and for any w-increasing substitutable choice functions F,G : 2^E → 2^E, there is a subset X of E such that

|X∩K|= 1 holds for any F G-kernel K. (3.1) Proof. For element xof E and subset Y of E define

Kx := ^

F{K ∈ KF G :x∈K}, K^x := _

F{K ∈ KF G :x∈K}, and K_Y := ^

F{K ∈ K_{F G} :K∩Y =∅}

and observe from Lemma 3.6 and (1.4) that

{K ∈ K_{F G} :x∈K}={K ∈ K_{F G} :K_{x F} K _F K^x}= [K_x, K^x], (3.2) that is, the set of F G-kernels that contain element x of E are exactly the ones in the interval between K_x and K^x. Consequently, our task is to partition the lattice of F G-kernels into disjoint intervals I_x = [K_x, K^x].

Pick an arbitrary element x₁ of F-optimal F G-kernel K¹ = KF and let I_x1 be the first interval in our partition. We create the intervals one by one, so assume that for somenwe have elementsx₁, x₂, . . . , x_nofEsuch that for anyi≤nwe have the following properties:

IntervalsIx1, Ix2, . . . , Ixn are pairwise disjoint and (3.3) K^x1 ≺F K^x2 ≺_F . . .≺_F K^xn and (3.4) Si

j=1I_xj = [KF, K^xi], that is, intervals I_xj cover

the part of the lattice of F G-kernels below K^xi. (3.5) Case 1. If x_n ∈ KG, i.e., if K^xn = KG then X = {x₁, x₂, . . . , x_n} has the property that Theorem 3.7 requires, and the proof is done.

CHAPTER 3. THE STRUCTURE OF KERNELS 28 Case 2. Otherwise, ifx_n6∈KGthen defineK :=K_{x

1,x2,...,xn}. AsK∩{x₁, x₂, . . . , x_n}=

∅ by (3.2) and the definition of K, (3.5) for i=n implies K 6_F K^xn. Clearly, K =_

F

{K_x :x∈K}, (3.6)

hence there must exist some x_n+1 ∈K such that

K_xn+1 6F K^xn . (3.7)

Thus K_xn+1 ∩ {x₁, x₂, . . . , x_n} = ∅ holds by (3.5) and K F K_xn+1 F K follows from (3.6) and the definition of K. This proves that

K =K_{x

1,x2,...,xn} =K_xn+1 (3.8)

Our next goal is to prove that (3.3) and (3.4) hold for n + 1 and (3.5) holds for i=n+ 1. To show (3.3), assume indirectlyK ∈I_xn+1∩I_xj,for some j ≤n. This means that K_xn+1 F K F K^xj F K^xn, contradicting (3.7). This contradiction proves (3.3) for n+ 1.

For (3.4), it is enough to justify K^xn F K^xn+1. As

χ(Kxn+1) +χ(K^xn) = χ(Kxn+1 ∧F K^xn) +χ(Kxn+1∧F K^xn)

holds by (1.4), and x_n+1 6∈K_xn+1∧F K^xn by the definition of K_xn and by (3.7). Conse-quently, x_n+1 ∈K_xn+1 ∨F K^xn and hence

K^xn F Kxn+1∨F K^xn F K^xn+1, proving (3.4) for n+ 1.

To show (3.5) for i = n+ 1, assume that K⁰ K^xn+1 holds for some F G-kernel K.

If x_n+1 6∈ K⁰ then K⁰ 6_F K_xn+1, hence K⁰ 6_F K = K_{x

1,x2,...,xn} due to (3.8). Hence K⁰∩ {x₁, x₂, . . . , x_n} 6=∅ and (3.5) for i=n+ 1 follows.

To finish the proof, we construct elementsx1, x2, . . .one after another as described in Case 2. above. As KF G is finite, at some point we end up in Case 1, and this completes the proof of Theorem 3.7.

Chapter 4

Applications of kernels

This chapter illustrates some applications of well-known theorems on stable matchings and less known results on kernels.

Below, we briefly sketch such an application. It is relatively easy to see that the Cantor-Bernstein theorem on cardinalities (stating that |A| ≤ |B|and |B| ≤ |A|implies

|A|=|B|) can be deduced from an infinite version of the stable matching theorem that we did not state in this work. Note that the Cantor-Bernstein theorem is a standard application of Theorem1.9of Tarski. Note also that the infinite stable matching theorem can be proved from Tarski’s fixed point theorem similarly as we proved the finite version due to Gale and Shapley. It is well-known (and easy to see) that the Cantor-Bernstein theorem can be regarded as an infinite version of the Mendelsohn-Dulmage theorem stating that if some matching covers subset A⁰ of partA of bipartite graph G and some other matching ofGcoversB⁰ of partB then some matching ofGcoversA∪B. From this perspective, it is not surprising that the Mendelsohn-Dulmage result can also be proved with stable matchings. Furthermore, the matroid generalization of the Mendelsohn-Dulmage theorem, namely the Kundu-Lawler theorem follows easily from Theorem 2.9, the matroid-generalization of Theorem 1.1 of Gale and Shapley.

In what follows, we survey kernel-related results about graph paths, graph colorings and college admissions.

4.1 Applications on paths

As we mentioned earlier, the graph-kernel result below can be proved from Tarski’s fixed point theorem (and also from Theorem 2.2).

Theorem 4.1 (Sands, Saurer, Woodrow [62]). If E1 and E2 are two loopless arc sets on vertex set V then there is a subset U of V with the two properties below.

• There is no directed path in E₁ or in E₂ connecting two different vertices of U and

• from any vertex v ∈V\U there exists a directed path ofE1 or ofE2 that terminates in U.

A perhaps less self-explanatory neat application of the theorem of Gale and Shapley is the proof of Pym’s theorem below.

CHAPTER 4. APPLICATIONS OF KERNELS 30 Theorem 4.2 (Pym [56]). Let each of P and Q be a set of vertex-disjoint directed paths in digraph D = (V, E). Then there exists a set R of vertex-disjoint directed paths of D such that

• there is a path of R starting at each starting vertex of a path of P and each path of R starts in a starting point of a path of P or of Q and

• any terminal of any path of Q is a terminal of some path of R and each terminal of each path of R is a terminal of a path of P or of Q, furthermore

• each path of R is a concatenation of a (possibly empty) starting segment of a path of P and a (possibly empty) end segment of a path of Q.

Proof. Define bipartite graph G on vertex set P ∪ Q such that edges correspond to common vertices of a path P ∈ P and Q∈ Q. For vertex P ∈ P define linear order ≤_P on E(P) according to the order on P of the intersection vertices of P. Similarly, linear order ≤_Q for path Q ∈ Q comes from the opposite order of the intersection vertices of Q. Let S be the intersection vertices of a stable matching ofG. DefineR as those paths of P ∪ Qthat do not contain any vertex of S and for each vertex v of S construct path R_v of R by concatenating the starting segment of a P path terminating at v with the terminal segment of a Q-path starting at v. Clearly, all three requirements for R hold, the only thing we have to check is that paths of R are vertex-disjoint.

Let v be an arbitrary vertex of D. As edge e_v of G that corresponds to v does not block stable matching S, e_v must be dominated by some edge e_u of S such that u and v are on the same path Z ∈ P ∪ Q. If Z belongs to P then the starting part of Z that belongs to R does not contain v and if Z ∈ Q then the terminal segment of Z that belongs to R does not containv. Hence no vertex v can belong to two paths ofR, that is, R consists of vertex-disjoint paths, as we claimed. This observation finishes the proof.