combinatorial topology in games and optimization { ojd

Combinatorial topology in games andoptimization – OJD

Frederic Meunier

May 25, 2015

Contents

1 Preliminaries 51.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Linear programming . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . 51.3.2 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . 71.4.2 Subgraphs . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.3 Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.4 Directed graphs . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Simplicial complexes . . . . . . . . . . . . . . . . . . . . . . . 101.7 Barycentric subdivision and posets . . . . . . . . . . . . . . . 121.8 Signed vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Sperner’s lemma, Tucker’s lemma, and their relatives 152.1 Sperner’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Ky Fan’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 The general Ky Fan’s lemma . . . . . . . . . . . . . . . 162.2.2 Ky Fan’s lemma for signed vectors . . . . . . . . . . . 17

3 Kneser graphs 193.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Fractional chromatic number . . . . . . . . . . . . . . . . . . . 193.3 Chromatic number . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Fair divisions 214.1 Cake cutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3

CONTENTS 4

4.1.1 Algorithmic features . . . . . . . . . . . . . . . . . . . 224.2 Necklace splitting . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2.1 Algorithmic features . . . . . . . . . . . . . . . . . . . 244.2.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . 24

5 Kernels in graphs 255.1 Combinatorial games . . . . . . . . . . . . . . . . . . . . . . . 255.2 Stable matchings . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 255.2.2 Bipartite case . . . . . . . . . . . . . . . . . . . . . . . 255.2.3 General case - HORS PROGRAMME . . . . . . . . . . 26

5.3 Kernels in graphs . . . . . . . . . . . . . . . . . . . . . . . . . 265.3.1 Definition and properties . . . . . . . . . . . . . . . . . 265.3.2 Fractional kernels . . . . . . . . . . . . . . . . . . . . . 27

5.4 Kernels in perfect graphs . . . . . . . . . . . . . . . . . . . . . 28

6 Nash equilibrium in bimatrix games 316.1 Nash equilibrium in bimatrix games . . . . . . . . . . . . . . . 316.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

CHAPTER 1

Preliminaries

1.1 Notations

For any positive integer a, the set {1, . . . , a} is denoted [a]. Let X be anyset. Given a nonnegative integer k, we denote by

(Xk

)the set of all subsets

of cardinality k of X.

1.2 Complexity

A problem is polynomial, or in P , if there exists a polynomial time algorithmsolving it.

A problem is NP -hard if unless P 6= NP , there is no polynomial algo-rithm solving it. The equality P = NP is unlikely. Being NP -hard can thusbe thought as being intrinsically hard.

1.3 Linear programming

1.3.1 Main results

A linear program is a mathematical program that can be written as

min cTxs.t. Ax ≥ b

x ∈ Rn.

where b ∈ Rm, c ∈ Rn, and A is a m× n real matrix. This is the inequationform. It can equivalently be written under the standard form

min cTxs.t. Ax = b

x ∈ Rn+

5

1.3. LINEAR PROGRAMMING 6

or the canonical formmin cTxs.t. Ax ≥ b

x ∈ Rn+.

The set of feasible solutions of a linear program is a polyhedron.

Theorem 1.3.1. Consider a minimization linear program.If it is feasible and bounded from below, then it has an optimal solution.If it has an optimal solution, then it has an optimal solution that is a

vertex of the polyhedron.If it is feasible but is not bounded from below, then there is a ray t 7→

x(t) = x0 + tq, for some x0, q ∈ Rn, with x(t) feasible for all t ∈ R+ suchthat limt→+∞ c

Tx(t) = −∞.

Proposition 1.3.2. Consider a linear program in any form such that b andA have only rational coefficients. If it admis an optimal solution, then it hasan optimal solution with rational coefficients.

Proof. This is a consequence of the second point of Theorem 1.3.1.

Note that there is no conditions on the coefficients of c in Proposi-tion 1.3.2.

1.3.2 Bases

Consider the systemAx = bx ≥ 0.

(P)

A is an m × n matrix of rank m. Given a subset X of [n], we denote byAX the matrix obtained by keeping from A only the columns indexed byelements of X.

A subset B of [n] is a basis if AB is nonsingular. Note that it implies that|B| = m. A basis is feasible if A−1B b is nonnegative. To each basis, there isan associated basic solution y defined by yB = A−1B b and y[n]\B = 0. Notethat if the basis is feasible, then the associated basic solution is feasible.

The pair (A, b) is generic if for any B ∈([n]m

), there is no y ∈ RB

+ suchthat ABy = b and y has a component equal to zero.

Lemma 1.3.3. Suppose (A, b) generic and {x ∈ Rn+ : Ax = b} bounded.

Let B ∈([n]m

). If ABy = b has a solution y in RB

+, then AB is nonsingular(i.e. B is a feasible basis).

7 CHAPTER 1. PRELIMINARIES

Proof. Suppose for a contradiction that there is a λ ∈ RB+ \ {0} such that

ABλ = 0. Then, for any t ∈ R+, we have AB(y + tλ) = b. Since {x ∈ Rn+ :

Ax = b} is bounded, there is necessarily a t such that y + tλ has at leastone component equal to 0, which contradicts the genericity assumption.

Proposition 1.3.4. Suppose (A, b) generic and {x ∈ Rn+ : Ax = b}

bounded. Let L ∈(

[n]m+1

). The number of feasible bases contained in L is

either 0 or 2.

Proof. Let Q = {y ∈ RL+ : ALy = b} and let λ ∈ KerAL\{0}. Assume that

there is a feasible basis B in L. We are going to show that there is exactlyone feasible basis in L distinct from B.

Let x ∈ R+B be defined by xB = A−1B b. Let z ∈ R+

L be defined by zi = xifor i ∈ B and z` = 0, where ` is the unique element in L \ B. Note thatwe have λ` 6= 0, otherwise AB would have been singular. Without loss ofgenerality, we assume that λ` > 0. Defining L to be the set of vectors z+ tλfor t ∈ R, we have Q = L ∩ RL

+. Since λ` > 0, if z + tλ ∈ Q, then t ≥ 0.When t goes to infinity, z + tλ leaves Q for some value α because of theboundness assumption. Thus, we have Q = {z + tλ : t ∈ [0, α]}.

Therefore, there are exactly two elements in Q having a support of sizem: the one obtained for t = 0 and the one obtained for t = α. Lemma 1.3.3allows to conclude.

1.4 Graphs

1.4.1 Basic definitions

A (undirected) graph is a pair G = (V,E), where V is a finite set and Ea finite family of unordered pairs uv from V . The elements of V are thevertices and the elements of E are the edges. We use the word ‘family’ ratherthan ‘set’ for the edges to indicate that some pairs of vertices may occurmore than once as an edge

Two vertices u and v are adjacent if uv is an edge. In such a case, u is aneighbor of v and vice-versa. The set of all neighbors of a vertex u is denotedN(u). An edge uv is incident to both u and v. The set of edges incident toa vertex v is denoted δ(v). The degree of a vertex v, denoted deg(v), is thenumber of edges incident to it: deg(v) = |δ(v)|.

The following lemma, whose proof is left in exercise, will be useful.

Lemma 1.4.1. The number of odd degree vertices in a graph is always even.

1.4. GRAPHS 8

A path is a sequence v0, e1, v1, e2, v2, . . . , e`, v` where the vi’s are vertices,where the ei’s are edges, and where ei = vi−1vi for all i. A circuit is a pathsuch that v0 = v`. A path (resp. circuit) is elementary if all ei are distinct.

A graph is connected if there is a path between all pairs of vertices.

A subset S of vertices is stable if no two vertices in S are adjacent. Thesize of a largest stable set is denoted α(G). Determining its value is anNP -hard problem.

G is complete if for all u, v ∈ V with u 6= v we have uv ∈ E (all possibleedges are present).

1.4.2 Subgraphs

A graph H = (V ′, E ′) is a subgraph of G = (V,E) if V ′ ⊆ V and E ′ ⊆ E.Given a subset X of V , we define H[X] to be the graph (X,E[X]) whereE[X] is the set of all edges of E having both endpoints in X. Such a graphis a subgraph of G and is induced by X.

A clique is a complete subgraph. The size of the largest clique in G isdenoted ω(G). Computing this quantity is an NP -hard problem.

1.4.3 Coloring

A coloring of a graph G is a map c : V → Z+. The elements in c(V ) are thecolors. The coloring is proper if no adjacent vertices get the same color. Aproper coloring can also be seen as a partition of the vertex set into stablesets. The chromatic number is the smallest number k such that there is aproper coloring with k colors. It is denoted χ(G). Determining χ(G) is anNP -hard problem.

An (a, b)-coloring of G is a family of a stable sets such that each vertexis covered at least b times. Since it is a family, the same stable set can occurseveral times in an (a, b)-coloring. The fractional chromatic number, denotedχf (G), is the quantity

inf{ab

: there exists a (a, b)-coloring of G}.

This infimum is actually a minimum as we are going to see. Since a propercoloring with a colors is an (a, 1)-coloring, we have

χf (G) ≤ χ(G).

Alternatively, the fractional chromatic number of G is the optimal value


of the following linear program:

min∑S∈S

xS

s.t.∑S∈Sv

xS ≥ 1 ∀v ∈ V

xS ∈ R+ ∀S ∈ S,

where S is the set of all stable sets of G and Sv is the set of those containingv. The first point of Theorem 1.3.1 explains then why there is an (a, b)-coloring achieving the fractional chromatic number and thus why the infimumis actually a minimum in the definition of the fractional chromatic number.

There is an easy lower bound on the fractional chromatic number.

Proposition 1.4.2.|V |α(G)

≤ χf (G).

Proof. Let F be an (a, b)-coloring. Denote by Fv the family of the stablesets in F containing v. We have

b|V | ≤∑v∈V

|Fv| =∑S∈F

|S| ≤ aα(G).

1.4.4 Directed graphs

A directed graph is a pair D = (V,A), where V is a finite set and A a finitefamily of ordered pairs (u, v) from V . The elements of V are the vertices andthe elements of A are the arcs.

Two vertices u and v are adjacent if (u, v) is an arc. In such a case, uis an inneighbor of v and v an outneighbor of u. The set of all outneighbors(resp. inneighbors) of a vertex u is denoted N+(u) (resp. N−(u)). An arc(u, v) leaves u and enters v. The set of arcs leaving (resp. entering) a vertexv is denoted δ+(v) (resp. δ−(v)). The outdegree (resp. indegree) of a vertexv, denoted deg+(v) (resp. deg−(v)), is the quantity |δ+(v)| (resp. |δ−(v)|).

A path is a sequence v0, a1, v1, a2, v2, . . . , a`, v` where the vi’s are vertices,where the ai’s are arcs, and where ai = (vi−1, vi) for all i. A cycle is a pathsuch that v0 = v`. A path (resp. cycle) is elementary if all ai are distinct.

A directed graph is strongly connected if for all u, v ∈ V , there is a pathfrom u to v and a path from v to u.

1.5. POLYHEDRA 10

d = 0 d = 1 d = 2 d = 3

Figure 1.1: d-dimensional simplices for d = 0, 1, 2, 3

1.5 Polyhedra

A polyhedron P of Rd is the intersection of finitely many closed halfspaces.A point v of P is a vertex, or an extreme point, of P if any segment of Pcontaining v in its interior is v itself. The set of all vertices is denoted V (P ).The intersection of the boundary of a polyhedron with a hyperplan avoidingits relative interior is a face. A face is also a polyhedron. A facet is a faceof maximal dimension. Note that a vertex is a 0-dimensional face. ∅ is theunique face of dimension −1.

A polytope is a bounded polyhedron. It can also be described as theconvex hull of finitely many points. In particular, any polytope is the convexhull of its vertices. A simplex is the convex hull of affinely independentpoints. It is a polytope. By definition, all faces of a simplex are againsimplices. The dimension of a simplex is also the number of its verticesminus one. Simplices in dimension 0, 1, 2, and 3 are depicted in Figure 1.1.For these values, the simplices have special names: a 1-dimensional simplexin an edge, a 2-dimensional simplex is a triangle, and a 3-dimensional simplexis a tetrahedron.

A polyhedral cone is a polyhedron C such that for any y ∈ C and anyλ ∈ R+, we have λy ∈ C. The extreme rays of C are the ray originating at0, contained in C, and not convex combination of any other rays in C. Forany polyhedron P , there exists a polytope Q and a polyhedral cone C suchthat any x ∈ P can be written as y+z, where y ∈ Q and z ∈ C. This is theMinkowski-Weyl theorem. Such a polyhedral cone C is uniquely determinedby P (it is not the case for the polytope Q). The extreme rays of P are theextreme rays of C.

1.6 Simplicial complexes

A simplicial complex K is a collection of simplices satisfying the followingproperties:


Figure 1.2: A simplicial complex

• if τ is a face of a simplex σ ∈ K, then τ ∈ K.

• if σ and σ′ are two simplices of K, then their intersection σ ∩ σ′ is aface of both (in particular, this intersection can be empty).

Its dimension – denoted dimK – is maxσ∈K dimσ. Its vertex set is theset of its 0-dimensional simplices: V (K) =

⋃σ∈K V (σ). The set of its edges

is denoted E(K) and is the set of its 1-dimensional simplices.

Figure 1.2 shows a simplex of dimension 2, whereas Figure 1.3 shows acollection of simplices that do not provide a simplicial complex (the intersec-tion of the two triangles is a face of none of them).

The union⋃σ∈K σ of all simplices of a simplicial complex K is called the

polyhedron of the simplicial complex. It is denoted ||K||.

1.7. BARYCENTRIC SUBDIVISION AND POSETS 12

Figure 1.3: Not a simplicial complex

Let X be a topological space. A simplicial complex T is a triangulationof X if ||T|| is homeomorphic to X.

1.7 Barycentric subdivision and posets

The barycentric subdivision of a simplicial complex K is defined as

sd(K) = {conv(bσ0 , . . . , bσk) : σ0, . . . , σk ∈ K \ {∅}, σ0 ⊂ · · · ⊂ σk},

where bσ denotes the barycenter of a simplex σ. The fact that sd(K) is indeeda simplicial complex deserves a proof.

Proposition 1.7.1. sd(K) is a simplicial complex.

Proof. To be completed

The following proposition, whose proof is left in exercise, is useful toobtain triangulation of arbitrarily small mesh.

Proposition 1.7.2.

diam(sdK) <dimK

1 + dimKdiam(K).

1.8 Signed vectors

A useful tool in combinatorics is the signed vectors. A signed vector is anelement of {+,−, 0}n for some positive n.

The partial order given by

0 ≺ +, 0 ≺ −, and + and − not comparable


is extended on signed vectors by taking the product order.

Given a signed vector x = (x1, . . . , xn), we denote by x+ (resp. x−) theset of indices such that xi = + (xi = −). We always have x+ ∩ x− = ∅.

1.8. SIGNED VECTORS 14

CHAPTER 2

Sperner’s lemma, Tucker’s lemma, andtheir relatives

2.1 Sperner’s lemma

Sperner’s lemma is an important result in combinatorial topology. It wasoriginally proposed by Sperner Sperner (1928) to obtain a simple and con-structive proof of Brouwer’s fixed-point theorem stating that any continuousmap from a finite-dimensional ball into itself has a fixed-point. Brouwer’sfixed-point theorem has numerous applications in mathematics and economy.The relation between Sperner’s lemma and Brouwer’s theorem can be foundfor instance in the recent book by De Longueville De Longueville (2012). Theoriginal proof by Sperner, even constructive, was not algorithmic. Motivatedby concrete applications, Scarf, inspired by the Lemke algorithm, proposedlater an algorithmic proof Scarf (1982), which is actually a pivot-based one.It is an adaptation of his proof that is given below.

Sperner’s lemma itself has many applications, in game theory or in com-binatorics. The application to cake cutting developed below is such an ex-ample. Another example is a result by Aharoni and Haxell Aharoni andHaxell (2000), who prove a very general sufficient condition for the existenceof a system of distinct representatives for a family of hypergraphs. Anothernice application of the Sperner lemma is a theorem due to Monsky Monsky(1970): any dissection of a rectangle into triangles of same area needs aneven number of triangles.

One of the multiple versions of Sperner’s lemma is the following theorem,proposed by Scarf Scarf (1967).

Theorem 2.1.1 (Sperner’s lemma). Let T be a triangulation of the standardd-dimensional simplex 4d. Let λ : V (T) → {0, 1, . . . , d} be a labeling of thevertices of T such that

• each vertex of 4d gets a distinct label,

15

2.2. KY FAN’S LEMMA 16

• each vertex v of T gets a label in λ(V (F )), where F is the minimal faceof 4d containing v.

Then there is an odd number of simplices σ ∈ T such that λ(V (σ)) ={0, 1, . . . , d}.

A simplex σ of T is fully labeled if λ(σ) = {0, 1, . . . , d}. Sperner’s lemmastates that under the conditions given on the labeling, there exists always anodd number of fully-labeled simplices.

Proof of Theorem 2.1.1. The proof works by induction on d. If d = 0, thereis nothing to prove.

If d > 0, we build a graph G whose vertices are the d-dimensionalsimplices, with an additional “dummy” vertex r. Two vertices are con-nected by an edge if the corresponding d-simplices share a facet τ such thatλ(V (τ)) = {0, . . . , d− 1}. Moreover, a vertex v of G is connected to r if thesimplex corresponding to v has a facet τ on the boundary of 4d such thatλ(V (τ)) = {0, . . . , d− 1}.

The degree of a vertex v 6= r is different from 0 only if the correspondingsimplex has a facet τ such that λ(V (τ)) = {0, . . . , d − 1}. This degree isalways 2 except if the simplex is fully-labeled, in which case the degree is1. By induction applied on the facet of 4d using labels 0, . . . , d − 1,weget that the degree of r is odd. There is thus another odd degree vertex(Lemma 1.4.1), which necessarily corresponds to a fully-labeled simplex.

Sperner’s lemma can be used to prove Brouwer’s fixed-point theorem.

2.2 Ky Fan’s lemma

2.2.1 The general Ky Fan’s lemma

The following theorem is due to Ky Fan Fan (1952).

Theorem 2.2.1. Suppose that T is a triangulation of the d-dimensionalsphere Sd such that if σ ∈ T then −σ ∈ T.

Let λ : V (T) → {−1,+1, . . . ,−m,+m} be a labeling of the vertices suchthat

• λ(−v) = −λ(v) for each v ∈ V (T)

• λ(u) + λ(v) 6= 0 for each edge uv ∈ E(T).

17 CHAPTER 2. SPERNER’S LEMMA, TUCKER’S LEMMA, AND THEIR RELATIVES

Then there is an odd number of d-simplices σ of T such that λ(V (σ)) has theform

{−j0,+j1, . . . , (−1)d+1jd}

with j0 < j1 < · · · < jd. In particular, we have m ≥ d+ 1.

The special case with m = d, which states that in this case a map withthe required conditions cannot exist, is known as Tucker’s lemma. Tuckerproved its special case for d = 2 in 1946 Tucker (1946). Tucker’s lemma canbe used to prove the Borsuk-Ulam theorem.

To author’s knowledge, there is no combinatorial proof for this version.We propose a proof for a triangulation with an additional property we de-scribe now.

We see Sd as being {(x1, . . . , xd+1) ∈ Rd+1 :∑d+1

i=1 x2i = 1} and decompose

it into hemispheres: H+0 , H

−0 , H

+1 , H

−1 , . . . , H

+d , H

−d . The hemisphere H+

d isthe set of points in Sd with xd+1 ≥ 0 and the hemisphere H−d is the set ofpoints in Sd with xd+1 ≤ 0. We recursively define the other hemispheressimilarly on the equator of Sd. The additional property we require for thetriangulation is that it induces a triangulation of the hemispheres (if theinterior of a simplex of T iintersects one of the hemispheres, then it is fullycontained in the hemisphere).

Proof of Theorem 2.2.1 for the special triangulation. We proceed by induc-tion on d. The theorem is obviously true for d = 0. Let us now suppose thatd > 0. For ε ∈ {−,+}, we say that a k-simplex σ is ε-alternating if λ(V (σ))has the form {εj0,−εj1, . . . , (−1)kεjk} with j0 < j1 < · · · < jk. We say thatit is almost ε-alternating it it has an ε-alternating facet without being itselfε-alternating.

We build a graph whose vertices are all d-simplices of H+d , with an ad-

ditional dummy vertex r. An edge links two vertices if the correspondingsimplices share a common alternating facet. There is also an edge betweenr and another vertex if this latter corresponds to a d-simplex having an −-alternating facet on the boundary of H+

d . By induction, the degree of r isodd. There is thus an additional odd number of odd degree vertices in thisgraph. They are exactly the alternating d-simplices of H+

d . There is thus anodd number of alternating simplices in H+

d . Using the symmetry, there is anodd number of −-alternating simplices on Sd.

2.2.2 Ky Fan’s lemma for signed vectors

The following theorem is a corollary of Theorem 2.2.1.

2.2. KY FAN’S LEMMA 18

Theorem 2.2.2. Let λ : {+,−, 0}n \ {(0, . . . , 0)} → {−1,+1, . . . ,−m,+m}be a map satisfying the following two properties:

• λ(−x) = −λ(x) for all x

• λ(x) + λ(y) 6= 0 for all x � y.

There there is an odd number of chains

x(1) ≺ x(2) ≺ · · · ≺ x(n)

such that λ({x(1), . . . ,x(n)}) has the form

{−j1,+j2, . . . , (−1)njn}

with j1 < j2 < · · · < jn. In particular, we have m ≥ n.

Proof. There is a one-to-one correspondence between signed vectors and thebarycenters of the proper faces of the (d+1)-dimensional cube. It is actuallya one-to-one correspondence between the chains and the simplices of thebarycentric subdivision of the boundary of the (d+ 1)-dimensional cube.

CHAPTER 3

Kneser graphs

3.1 Introduction

For positive integers n, k such that n ≥ 2k, we define the Kneser graphKG(n, k) = (V,E) by

V =([n]k

)E = {{e, f} : e, f ∈ V, e ∩ f = ∅}.

Kneser graphs have been extensively studied, see ?? among many refer-ences. Martine Kneser ? conjectured the following result, which was finallyconfirmed by Lovasz’s theorem ?.

Theorem 3.1.1 (Lovasz theorem). Given n and k two positive integers withn ≥ 2k, we have χ(KG(n, k)) = n− 2k + 2.

Lovasz actually proved more than this theorem and gave a general lowerbound on the chromatic number of a graph in terms of the connectivity ofa simplicial complex associated to the graph. One of his main tool was theBorsuk-Ulam theorem and his paper is often seen as the first application ofalgebraic topology to combinatorics.

Kneser graphs have received a great attention not only because of thebeauty of Kneser’s conjecture but also because they enjoy many interestingproperties. One of them is an arbitrary gap between their fractional chro-matic number and the chromatic number.

3.2 Fractional chromatic number

Theorem 3.2.1. Given n and k two positive integers with n ≥ 2k, we have

χf (KG(n, k)) =n

k.


19

3.3. CHROMATIC NUMBER 20

3.3 Chromatic number

The fact that χ(KG(n, k)) ≤ n − 2k + 2 was already known by Kneser andcan be proved by the following explicit coloring

c : A ∈(

[n]

k

)→ min(minA, n− 2k + 2) ∈ [n− 2k + 2].

Proof of Theorem 3.1.1. It is enough to prove that χ(KG(n, k)) ≥ n−2k+2.Let c be a proper coloring of KG(n, k) with t colors.

Let m = t + 2k − 2. We define a map λ : {+,−, 0}n \ {(0, . . . , 0)} →{−1,+1, . . . ,−m,+m} as follows. Let x ∈ {+,−, 0}n \ {(0, . . . , 0)}.

If |x+| + |x−| ≤ 2k − 2, we define λ(x) to be ε(|x+| + |x−|), whereε = mini∈[n]{xi : xi 6= 0} (first nonzero entry of x).

If |x+| + |x−| ≥ 2k − 1, we define λ(x) to be ε(2k − 2 + min(a+, a−)),where

a+ = min{c(S) : S ⊆ x+ and |S| = k} and a− = min{c(S) : S ⊆ x− and |S| = k}

and where ε = + if a+ < a− and ε = − otherwise.

It can be checked that λ satisfies the condition of Theorem 2.2.2. Wehave thus t+ 2k − 2 ≤ n, which implies the whished inequality.

CHAPTER 4

Fair divisions

In many situations, goods have to be divided between players in a “fair”way. Fair can have several different meanings. The division can be fair ifeach player has the feeling to have at least as much as any other player. Thedivision can also be fair when the division is made by a board of examinersand the division is fair according to each examiner’s evaluation. And so on.

In the present chapter, we present two families of result with distinct def-inition of fairness, namely cake-cutting and necklace-splitting. These resultsare fascinating by many aspects.

4.1 Cake cutting

In this section, we focus on the following problem: cut a cake in such a wayeach player has the feeling to get at least as much as any other player. Wewill see that under natural assumption, it is possible to cut a cake among nplayers with n− 1 cuts and get such a division, which is said to be envy-free.It is a theorem due to Stromquist. The main tool for proving this result isSperner’s lemma.

According to Su Su (1999), Forest Simmons found a constructive proofof Stromquist’s theorem based on Sperner’s lemma Sperner (1928), the com-binatorial counterpart of Brouwer’s fixed point theorem.

Each cake is identified with the interval [0, 1]. A division of the cakeinto r pieces is an r-tuple x = (x1, . . . , xr), with xj ≥ 0 for all j ∈ [r]and

∑rj=1 xj = 1, where xj is the size of the jth piece (ordered from left

to right) of the cake. Given a division x of the cake into r pieces, a playerprefers a certain piece if that player does not think that any other piece isstrictly better. For some divisions a player may be indifferent to two or more“preferred” pieces.

The following assumptions on the preferences are the ones considered byStromquist.

1. Independence of preferences: The preferences of one player do not de-

21

4.1. CAKE CUTTING 22

pend of the choices made by the other players.

2. The players are hungry: A player will never choose an empty piece.

3. Preference sets are closed: If one player prefer the same piece for aconvergent sequence of division, then that piece will be preferred atthe limit.

Theorem 4.1.1. Assume that r players want to divide the cake.There alwaysexists an envy-free division into r pieces.

The proof is based on Sperner’s lemma and uses actually two distinctlabelings. Let T be a triangulation of the standard simplex 4r−1. A labelingµ : V (T)→ [r] is an owner-labeling if each (r−1)-simplex σ gets all labels onits vertices: µ(V (σ)) = [r]. Not all triangulations admit an owner-labeling.

Lemma 4.1.2. For all ε > 0, there exists a triangulation of the standardsimplex admitting an owner-labeling and such that diamT < ε.

Proof. Indeed, the barycentric subdivision of any triangulation of the stan-dard simplex admits an owner-labeling.

Proof of Theorem 4.1.1. For each point x ∈ 4r−1, we define νi(x) to be thepiece preferred by player i when the division of the cake is x (in case of atie, make an arbitrary choice).

We take T a triangulation of the standard simplex 4r−1 with an owner-labeling µ and we suppose T to be of arbitrary small mesh size. It existsaccording to Lemma 4.1.2. For each vertex v of T, we define λ(v) to beνµ(v)(x), where x is the coordinate of v in 4r−1. This provides a map λ :V (T) → [r]. Because of the assumption that the players are hungry, λ is aSperner labeling, and Theorem 2.1.1 applies. There is thus a fully-labeledsimplex.

Since the mesh size is taken arbitrarily small, the compactness of 4r−1

and the closedness of the preference sets imply that there is a point x∗ ∈ 4r−1

such that the νi(x∗)’s are all distinct. This point x∗ is the sought envy-free

division.

4.1.1 Algorithmic features

The proof of Sperner’s lemma (Theorem 2.1.1) can be made completely al-gorithmic. The algorithm underlying such a proof of Sperner’s lemma canbe used to compute, for any ε > 0, an “approximate” envy-free division.Tobe completed.

23 CHAPTER 4. FAIR DIVISIONS

4.2 Necklace splitting

Two thieves have stolen a beautiful necklace with n beads on a string madeof gold. Each bead is of a certain type, the total number of types being t.The thieves wish to divide the necklace, in such a way that each of themgets the same number of beads of each type. Assuming the number of beadsper type being even, such a division clearly exists, at least by cutting thenecklace between each pair of adjacent beads. We will see that a divisionwith this notion of fairness not requiring more that t cuts exists, whateverthe total number n of beads is. The following theorem was first proved byGoldberg and West Goldberg and West (1985) in 1985. A simpler proof usingthe Borsuk-Ulam theorem was proposed by Alon and West Alon and West(1986) the following year.

Denote by aj the number of beads of type j. A division of the necklaceis fair if for any bead type j, each thief gets at least baj

2c beads of type j.

Theorem 4.2.1 (Necklace theorem). It is possible to split fairly the necklacebetween the two thieves with no more than t cuts.

We give here the combinatorial proof proposed by Palvolgyi. A sequenceof elements in {+,−, 0} is alternating if there is no term equal to 0 and twoconsecutive terms are opposite. For x ∈ {+,−, 0}n, we denote by alt(x) thelongest alternating subsequence of x1, . . . , xn.

Proof of Theorem 4.2.1. We arbitrarily assign to one of the thieves the sign+ and to the other the sign −. For a contradiction, we assume that allfair splittings require at least t + 1 cuts. Let g(x) = max{alt(y) : y ∈{+,−, 0}n, y � x}. We define a map λ : {+,−, 0}n \ {(0, . . . , 0)} →{−1,+1, . . . ,−(n−1),+(n−1)} as follows. Let x ∈ {+,−, 0}n\{(0, . . . , 0)}.Notice that x encodes a splitting of the necklace with alt(x)− 1 cuts.

If g(x) ≥ t + 2, we define λ(x) to be ε(g(x) − 1), where ε = + if they for which the maximum is attained in the definition of g starts with a +,and ε = − otherwise. There is no ambiguity in the definition: there may beseveral y attaining the maximum, but they all start with the same sign.

If g(x) ≤ t + 1, we define λ(x) as follows. For each i, we give the beadsituated at position i to the + thief if xi = +, to the − thief if xi = −, andto none of the thieves if xi = 0. Suppose that for each type j, each thief getsat most aj/2 beads of this type. We can complete x in order to give to eachthief at least aj/2 beads of each type j. By definition of g(x), it is a fairsplitting with at most t cuts. Since we have supposed that such a splittingdoes not exist, it implies that there is at least one type j∗ such that one ofthe thief gets strictly more than aj∗/2 beads of this type. Choose such a j∗

4.2. NECKLACE SPLITTING 24

as small as possible, and define ε to be the sign of the thief getting strictlymore that aj∗/2 beads of type j∗. We define then λ(x) to be εj∗.

It can be checked that λ satisfies the condition of Theorem 2.2.2, which isimpossible since n is larger than n− 1. Thus the starting assumption cannotbe done and there is a splitting as wished.

4.2.1 Algorithmic features

4.2.2 Generalizations

CHAPTER 5

Kernels in graphs

5.1 Combinatorial games

Winning positions.

5.2 Stable matchings

5.2.1 Definition

Let G = (V,E) be a graph and assume that there is a total order �v on theedges in δ(v) for each vertex v in V . A matching M ⊆ E is said to be stableif for any edge e = uv not in M there is an edge f in M such that e ≺u for e ≺v f . A stable matching does not necessarily exist. For instance, if Gis a triangle with vertices u1, u2, u3 and if ui−1ui ≺ui uiui+1 for i = 1, 2, 3(with u4 := u1 and u0 := u3), there is no stable matching. However, if G isbipartite, a stable matching always exists.

5.2.2 Bipartite case

Stable matchings were originally defined for bipartite graphs by Gale andShapley Gale (1962). They proved the following theorem, actually with aslightly different, but clearly more appealing, formulation. In their formula-tion, one part of the bipartite graph is the set of women, while the other isthe set of men. Each woman would agree to marry with some men, and isable to rank them. The same holds for men. A stable marriage is a stablematching in the bipartite graphs whose edges model possible marriage. [...]

Shapley got in 2012 the Nobel prize in economics for his work on coop-erative games, and in particular for this theorem.

Theorem 5.2.1 (Gale-Shapley theorem). If G is bipartite, a stable matchingexists.


25

5.3. KERNELS IN GRAPHS 26

The proof is constructive and provides a polynomial-time algorithm (inO(|V ||E|)) for computing a stable matching.

Application: the Galvin theorem for list-coloring

The Gale-Shapley theorem has a nice application in the theory of graph edge-coloring, namely Galvin’s theorem Galvin (1995), which solves the Dinitzconjecture, cited in Erdos et al. (1979).

5.2.3 General case - HORS PROGRAMME

The following theorem, found by Tan Tan (1991) in 1991, is a generalizationof Theorem 5.2.1. A sequence v1, e1, v2, e2, . . . , vk, ek with vi ∈ V and ei =vivi+1 for i = 1, . . . , k is a preference cycle if

• ei−1 ≺vi ei for i = 1, . . . , k

• ek ≺v1 e1.

The integer k is the length of the preference cycle.

Theorem 5.2.2 (Tan theorem). If there are no preference cycles of oddlength, a stable matching exists.

Again, the proof is constructive and provides a polynomial-time algo-rithm.

5.3 Kernels in graphs

5.3.1 Definition and properties

A kernel in a directed graph D = (V,A) is a subset K of the vertices such thatA[K] = ∅ and K ∪N−(K) = V . In other words, given a kernel K, any twovertices in K are nonadjacent and any vertex not in K has an outneighborin K.

We give now two examples of kernels. The set of winning positions inthe position graph of a combinatorial game forms a kernel, as noted by VonNeumann and Morgenstern Von Neumann and Morgenstern (1944). Stablematchings can be seen as a special case of kernels as follows. Given anundirected graph G = (V,E) and a family of total orders �v on the setsδ(v), we consider the directed graph D′ = (E,A′), whose vertex set is theedge set of G, and in which the arc (e, f) exists only if e �v f for a common

27 CHAPTER 5. KERNELS IN GRAPHS

endpoint v. The directed graph D′ is an orientation of the line-graph of G.The stable matchings of G coincide with the kernels of D′.

The following theorem is one of the most general existence result forkernels. It is due to Richardson Richardson (1953).

Theorem 5.3.1. If there are no odd directed cycles in D, then there existsa kernel.

The proof is constructive and a kernel can be found in polynomial timefor such a directed graph. Actually, as shown by a series of works byDuchet, Meyniel, Galeana-Sanches, Neumann-Lara Duchet (1980), Duchetand Meyniel (1983), Galeana-Sanches and Neumann-Lara (1984), if each odddirected cycle of D has two chords whose heads are two consecutive verticesof the cycle, then there is a kernel. Again, the proof is constructive andprovides a polynomial-time algorithm.

In general, deciding the existence of a kernel is difficult, as shown byChvatal Chvatal (1973).

Theorem 5.3.2. Deciding whether a directed graph has a kernel is NP -complete.

5.3.2 Fractional kernels

Let D = (V,A) be a directed graph. If x ∈ RV+ is such that

∑v∈C xv ≤ 1

for each clique C of D = (V,A) and such that∑

v∈N+(u)∪{u} xv ≥ 1 for eachvertex u, then x is a fractional kernel. It is a strong fractional kernel ifmoreover for each vertex u, there is a clique Cu in N+(u) ∪ {u} such that∑

v∈Cuxv ≥ 1. Note that the incidence vector of a kernel is a fractional

kernel, and even a strong fractional kernel.The main result of this section is the following theorem by Aharoni and

Holsman.

Theorem 5.3.3. If D has no length 3 directed cycles, then D admits a strongfractional kernel.

Note that the condition of not having any length 3 directed cycle is equiv-alent of each clique having a directed cycle of length at most 2.

The proof uses the following theorem by Kiraly and Pap.

Theorem 5.3.4. Let P be a pointed n-dimensional polyhedron with exactlyn independent extreme rays, numbered from 1 to n. Suppose that the facetsare colored with the colors [n]. If the facets containing the extreme ray i arenot colored with color i, then there exists an extreme point lying on facets ofevery color.

5.4. KERNELS IN PERFECT GRAPHS 28


Proof of Theorem 5.3.3. For each directed cycle of length 2, we arbitrarilyget rid of one of the two arcs. We can do this transformation without loss ofgenerality since any strong fractional kernel of this new directed graph is astrong fractional kernel of the starting graph.

Let n be the cardinality of V , and let us consider the polyhedron

P = {x ∈ RV :∑v∈C

xv ≤ 1 for each clique C}.

P is of dimension n since 0 is in its interior. P is pointed since the incidencevector of any maximal stable set is an extreme point of P . The extreme raysare the rays of directions −e(v), where the e(v) are the unit vectors of thestandard orthogonal basis, v ∈ V .

We first check that every extreme point of P has no negative coordinates.Assume for a contradiction that x = (xv)v∈V is an extreme point of P andthat xu < 0 for some vertex u. Since x is an extreme point, there is a cliqueC having u has a vertex and such that

∑v∈C xv = 1. Indeed, otherwise, we

could write x has a convex combination of two points of P differing only ontheir u component. But then, we have

∑v∈C\{u} xv > 1, which contradicts

the definition of P , since C \ {u} is again a clique.Let us now color the facets of P with V as color set as follows. Since D

has no length 3 directed cycles, each clique C has no cycles at all and hasthus a unique sink sC . Each facet is characterized by a clique C such that∑

v∈C xv = 1. We color such a facet with vertex sC . Such a coloring satisfiesthe condition of Theorem 5.3.4. There exists thus an extreme point y ∈ RV

+

lying on facets of every color. For each vertex u ∈ V , there is a clique Cusuch that sCu = u and

∑v∈Cu

yv = 1. Since by definition Cu ⊆ N+(u)∪{u},we get that y is a strong fractional kernel.

5.4 Kernels in perfect graphs

A perfect graph is a graph G = (V,E) such that ω(H) = χ(H) for all inducedsubgraphs H of G (the graph G itself included).

The following theorem, conjectured by Berge and Duchet in 1980, hasbeen proved by Boros and Gurvich in 1996.

Theorem 5.4.1. Let G be a perfect graph. Let us orient the edges of G issuch a way that there are no directed cycles of length 3. Then the resultingdigraph ~G has a kernel.

29 CHAPTER 5. KERNELS IN GRAPHS

The proof we are going to give is due to Kiraly and Pap. The proof isshorter than the original one, which was already simplified by Aharoni andHolsman. It uses the following proposition as a key tool for deducing thesought conclusion.

Proposition 5.4.2. Let (wv)v∈V be weights attached to the vertices of aperfect graph G. Then there exists a stable set intersecting each clique ofmaximal weight.


Proof of Theorem 5.4.1. Let x = (xv)v∈V be a strong fractional kernel of ~G.Let us consider the xv as weights on the vertices. The cliques of maximalweight are of weight equaling 1. According Proposition 5.4.2, there is a stableset K intersecting each clique of weight 1. For each vertex u, there is a cliqueCu in N+(u) ∪ {u} such that

∑v∈Cu

xv = 1. It implies that for each vertexu, there is vertex in K contained in N+(u)∪ {u}. The set K is a kernel.

5.4. KERNELS IN PERFECT GRAPHS 30

CHAPTER 6

Nash equilibrium in bimatrix games

6.1 Nash equilibrium in bimatrix games

Let A = (aij) and B = (bij) be two m × n matrices with real coefficients.They form the input of a bimatrix game: a first player selects an index i in[m], while a second player selects an index j in [n]. The payoff of first playeris aij while the payoff of the second player is bij. They both try to maximizetheir payoff.

A Nash equilibrium is a pair (i∗, j∗) such that

ai′j∗ ≤ ai∗j∗ for all i′ ∈ [m] and bi∗j′ ≤ bi∗j∗ for all j′ ∈ [n].

It models a situation in which no player is incited to modify his choice. Nashequilibrium does not necessarily exist. However, with the weaker notion ofmixed Nash equilibirum we explain now, existence is always ensured.

Let 4k be the set of vectors x ∈ Rk+ such that

∑ki=1 xi = 1. A mixed

Nash equilibrium is a pair (y∗, z∗) with y∗ ∈ 4n and z∗ ∈ 4m such that

zTAy∗ ≤ z∗TAy∗ for all z ∈ 4m and z∗TBy ≤ z∗TBy∗ for all y ∈ 4n.(6.1)

It models the situation when the players choose a probability distribution ontheir choice set and then let chance decides.

In such a situation, there is always a mixed Nash equilibrium. It is acorollary of the famous Nash theorem ?.

Theorem 6.1.1. A bimatrix game always admits a mixed Nash equilibrium.

6.2 Proof

The proof of Theorem 6.1.1 relies on the following proposition.

31

6.2. PROOF 32

Consider the two systems:[Im, A]x =

1...1

x ≥ 0

(6.2)

[BT , In]x =

1...1

x ≥ 0

(6.3)

Proposition 6.2.1. Assume that both systems (6.2) and (6.3) are genericand bounded. Then there exists an even number of pairs (U, V ) such that

• U is a feasible basis of system (6.2)

• V is a feasible basis of system (6.3)

• U ∩ V = ∅.

Note that the bases of system (6.2) are of cardinality m, while thoseof system (6.3) are of cardinality n. It implies that the last point in thestatement of the proposition can be restated as “U ∪ V = [m+ n]”.

Proof of Proposition 6.2.1. To be completed

Proof of Theorem 6.1.1. Adding a same value to all entries of A and B doesnot modify the game. We can thus assume that the systems (6.2) and (6.3)are bounded.

By a standard technique of linear programming, it is possible to perturbeA and B in such a way that the systems are also generic and such that anybasis of the perturbed systems is also a basis of the original systems. We canthus apply Proposition 6.2.1 on the perturbed systems. ([m], [m + n] \ [m])is a pair as in the statement of Proposition 6.2.1. There exists thus anothersuch pair (U , V ).

Define u to be the basic solution associated to U and v to be the oneassociated to V , for the original non-perturbated systems. We write u as(r, y), with r being the first m components of u and y being the last ncomponents of u. Similarly, we write v as (z, s), with z being the first mcomponents of z and s being the last n components of v. We define

y∗ =1

||y||1y and z∗ =

1

||z||1z.

33 CHAPTER 6. NASH EQUILIBRIUM IN BIMATRIX GAMES

This definition is possible since we have y 6= 0 and z 6= 0. This is due to thefact that U 6= [m] and V 6= [m+ n] 6= [m]. We claim that (y∗, z∗) is a Nashequilibrium. The remaining of the proof is devoted to this statement.

We have

r + Ay =

1...1

.

Multiplying on the left by any z ∈ 4m, we get

zTr + zTAy = 1.

Thus, we have

zTAy∗ ≤ 1

||y||1.

Multiplying on the left by z∗ gives

z∗TAy∗ =1

||y||1.

Indeed, the fact U ∩ V = ∅ implies that the supports of r and z∗ are disjoint.Therefore,

zTAy∗ ≤ z∗TAy∗ for all z ∈ 4m.

as required.The inequality involving B in the definition of the Nash equilibrium is

proved similarly.

6.2. PROOF 34

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