[PPT] - Coalition games on interaction graphs or a nice way to play with PowerPoint Presentation

SLIDE 1

Coalition games

n interaction graphs
r a nice way to play with treewidth and brambles.

Nicolas Bousquet

joint work with Zhentao Li and Adrian Vetta

Mars 2015

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SLIDE 2

Context

We want people to work together.

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SLIDE 3

Context

We want people to work together.
Unfortunately, people are selfish : if it is is more interesting

for them, they will create a project of their own.

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SLIDE 4

Context

We want people to work together.
Unfortunately, people are selfish : if it is is more interesting

for them, they will create a project of their own.

Solution : distribute payoff in such a way people do not want

to leave.

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SLIDE 5

Coalition games

A set I of n agents.
A valuation function v : 2n → N. (the gain of any group

deciding to work on its own project)

Coalition game A subset S of agents is a coalition if v(S) is positive. Definition (coalition)

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SLIDE 6

Coalition games

A set I of n agents.
A valuation function v : 2n → N. (the gain of any group

deciding to work on its own project)

Coalition game A subset S of agents is a coalition if v(S) is positive. Definition (coalition) Distribute money to the agents in such a way, for every coalition S, the money distributed to agents of S is at least v(S). ⇒ No one wants to leave the grand coalition and work on its

wn project.

Goal

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SLIDE 7

Core

The core of the coalition game is the set of vectors of payoff satisfying the following constraints :

i∈I xi = v(I)

and

i∈S xi ≥ v(S)

∀S ⊆ I xi ≥ 0 ∀i ∈ I Definition (core)

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SLIDE 8

Core

The core of the coalition game is the set of vectors of payoff satisfying the following constraints :

i∈I xi = v(I)

and

i∈S xi ≥ v(S)

∀S ⊆ I xi ≥ 0 ∀i ∈ I Definition (core) Problem : The core can be empty !

Which conditions ensure that the core is not empty ?
Weaken the definition of core.

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SLIDE 9

Core

The core of the coalition game is the set of vectors of payoff satisfying the following constraints :

i∈I xi = v(I)

and

i∈S xi ≥ v(S)

∀S ⊆ I xi ≥ 0 ∀i ∈ I Definition (core) Problem : The core can be empty !

Which conditions ensure that the core is not empty ?
Weaken the definition of core.

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SLIDE 10

Least core

Least-Core: max

α s.t.

i∈I

xi = v(I) and

i:i∈S

xi ≥ α · v(S) ∀S ⊆ I xi ≥ 0 Definition (multiplicative least core) Intuition : Leaving the grand coalition has a cost. Thus agents stay in the grand coalition unless they have huge profit if they left it.

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SLIDE 11

Relative cost of stability

Another approach (relative cost of stability) : How much money must be injected by an external authority to stabilize the system ? ⇒ Our expenses minus our gains.

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SLIDE 12

Relative cost of stability

Another approach (relative cost of stability) : How much money must be injected by an external authority to stabilize the system ? ⇒ Our expenses minus our gains. Our gains :

Packing-LP: ν(G) = max

S:S⊆I

v(S) · yS s.t.

S⊆I:i∈S

yS ≤ 1 ∀i ∈ I yS is integral and ≥ 0 ∀S ⊆ I If v is super-additive then the packing is of I is precisely v(I). ⇒ The (integral) packing represents the amount of money we gain.

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Our expenses

Covering-LP: τ(G)∗ = min

i∈I

xi s.t.

i:i∈S

xi ≥ v(S) ∀S ⊆ I xi ≥ 0 It represents the amount of money which has to be spent in order to stabilize the system.

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SLIDE 14

Our expenses

Covering-LP: τ(G)∗ = min

i∈I

xi s.t.

i:i∈S

xi ≥ v(S) ∀S ⊆ I xi ≥ 0 It represents the amount of money which has to be spent in order to stabilize the system. Simplification : ⇒ Since τ ∗ − ν is not “stable” (by disjoint copy for instance), we consider τ ∗

ν instead.

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SLIDE 15

Relative Cost of Stability

The relative cost of stability of a game G is the ratio τ ∗(G)

ν(G) .

Definition (relative cost of stability)

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SLIDE 16

Relative Cost of Stability

The relative cost of stability of a game G is the ratio τ ∗(G)

ν(G) .

Definition (relative cost of stability) Game theoretical interpretation : The relative cost of stability represents the ratio between the minimum payment satisfying the constraints and the total wealth the grand coalition can generate.

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SLIDE 17

Relative Cost of Stability

The relative cost of stability of a game G is the ratio τ ∗(G)

ν(G) .

Definition (relative cost of stability) Game theoretical interpretation : The relative cost of stability represents the ratio between the minimum payment satisfying the constraints and the total wealth the grand coalition can generate. By Strong Duality Theorem, we have : ν(G) ≤ ν∗(G) = τ ∗(G) ≤ τ(G) Thus τ ∗ ν = ν∗ ν ≤ τ ν

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Hypergraph representation

For simplicity, we will focus on simple games, i.e., games where each coalition has value 0 or 1.

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Hypergraph representation

For simplicity, we will focus on simple games, i.e., games where each coalition has value 0 or 1.

τ(G) = min

i∈I

xi s.t.

i:i∈S

xi ≥ v(S) ∀S ⊆ I

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SLIDE 20

Hypergraph representation

For simplicity, we will focus on simple games, i.e., games where each coalition has value 0 or 1.

τ(G) = min

i∈I

xi s.t.

i:i∈S

xi ≥ v(S) ∀S ⊆ I

ν(G) = max
S:S⊆I

v(S) · yS s.t.

S⊆I:i∈S

yS ≤ 1 ∀i ∈ I

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SLIDE 21

Worst case

The Packing-Covering ratio τ(G)

ν(G) (and both integrality gaps) can

be arbitrarily large. Lemma

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Worst case

The Packing-Covering ratio τ(G)

ν(G) (and both integrality gaps) can

be arbitrarily large. Lemma Example :

A set of agents I = {1, . . . , n}
A subset S of I is a coalition if and only if |S| > |I|/2.
Maximum Packing : 1.
Minimum Covering : ≥ |I|

2 − 1.

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SLIDE 23

Worst case

The Packing-Covering ratio τ(G)

ν(G) (and both integrality gaps) can

be arbitrarily large. Lemma Example :

A set of agents I = {1, . . . , n}
A subset S of I is a coalition if and only if |S| > |I|/2.
Maximum Packing : 1.
Minimum Covering : ≥ |I|

2 − 1.

Which conditions imply an Erd˝

s-Pos´

a property ? Question

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SLIDE 24

Interaction graph

Myerson proposed the following model : Let G be a graph where the vertices of G are the agents of the coalition game G. The game G has interaction graph G if every coalition is connec- ted (i.e., if v(S) > 0 then S is connected). Definition (interaction graph)

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Interaction graph

Myerson proposed the following model : Let G be a graph where the vertices of G are the agents of the coalition game G. The game G has interaction graph G if every coalition is connec- ted (i.e., if v(S) > 0 then S is connected). Definition (interaction graph) Motivation : The agents must be able to communicate if they want to create a coalition.

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SLIDE 26

Interaction graph

Myerson proposed the following model : Let G be a graph where the vertices of G are the agents of the coalition game G. The game G has interaction graph G if every coalition is connec- ted (i.e., if v(S) > 0 then S is connected). Definition (interaction graph) Motivation : The agents must be able to communicate if they want to create a coalition. Examples :

G is a clique : any coalition may exist.
G is a stable set : coalitions have size one.

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SLIDE 27

Treewidth and coalition game

Let G be a graph. We have the following inequality : τ(G) ν(G) ≤∀ tw(G) + 1 Moreover there exist graphs for which this bound is tight. Theorem (Meir et al.) By ≤∀, we mean that every game G on interaction graph G satisfies this inequality.

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Treewidth and coalition game

Let G be a graph. We have the following inequality : τ(G) ν(G) ≤∀ tw(G) + 1 Moreover there exist graphs for which this bound is tight. Theorem (Meir et al.) By ≤∀, we mean that every game G on interaction graph G satisfies this inequality. Our work : Improve this result, and look at the integrality gaps

ν∗(G) ν(G) and τ(G) τ ∗(G).

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Reminder : treewidth

A tree T and a (bag) function f : T → 2V is a tree decomposition

f G = (V , E) if :
For every v ∈ V , the set of nodes containing v in their bags is

a subtree Tv of T.

For every edge uv, Tu and Tv intersects.

The width of a decomposition is the maximum size of a bag of the tree-decomposition minus one. The treewidth of G, tw(G), is the minimum width of a tree- decomposition of G. Definition (treewidth)

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SLIDE 30

Reminder : treewidth

A tree T and a (bag) function f : T → 2V is a tree decomposition

f G = (V , E) if :
For every v ∈ V , the set of nodes containing v in their bags is

a subtree Tv of T.

For every edge uv, Tu and Tv intersects.

The width of a decomposition is the maximum size of a bag of the tree-decomposition minus one. The treewidth of G, tw(G), is the minimum width of a tree- decomposition of G. Definition (treewidth) The −1.

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Examples

Kn has a tree decomposition of width n − 1 (all the vertices

are in the same bag).

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Examples

Kn has a tree decomposition of width n − 1 (all the vertices

are in the same bag).

Trees have tree decompositions of width 1.

a b c d e f g h b, c b, d a, b a, e e, f e, g g, h

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Examples

Kn has a tree decomposition of width n − 1 (all the vertices

are in the same bag).

Trees have tree decompositions of width 1.
The d × d grid has a tree decomposition of width d.

a b c d e f g h b, c b, d a, b a, e e, f e, g g, h a b c d e f g h i a, b c, d b, c d, e c, d e, f · · · f, g h, i

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Dual notion : bramble

A set of vertices V1, . . . , Vℓ is a bramble of order k if

For every i, Vi is connected.
For every i = j, Vi and Vj intersect or share an edge.
The minimum number of vertices intersecting all the sets

V1, . . . , Vℓ is (k + 1). Definition (bramble)

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Dual notion : bramble

A set of vertices V1, . . . , Vℓ is a bramble of order k if

For every i, Vi is connected.
For every i = j, Vi and Vj intersect or share an edge.
The minimum number of vertices intersecting all the sets

V1, . . . , Vℓ is (k + 1). Definition (bramble) The treewidth of the graph G is equal to the maximum order of a bramble of G. Theorem (Robertson, Seymour)

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Dual notion : bramble

A set of vertices V1, . . . , Vℓ is a bramble of order k if

For every i, Vi is connected.
For every i = j, Vi and Vj intersect or share an edge.
The minimum number of vertices intersecting all the sets

V1, . . . , Vℓ is (k + 1). Definition (bramble) The treewidth of the graph G is equal to the maximum order of a bramble of G. Theorem (Robertson, Seymour) The +1 and the fact that the family does not intersect.

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Examples

A set of vertices V1, . . . , Vℓ is a bramble of order k if The sets V1, . . . , Vℓ are connected. The sets V1, . . . , Vℓ are pairwise intersecting or share an edge. k + 1 vertices are needed to hit V1, . . . , Vℓ.

Cliques of size k : a bramble of order (k − 1) where Vi = {i}.

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Examples

A set of vertices V1, . . . , Vℓ is a bramble of order k if The sets V1, . . . , Vℓ are connected. The sets V1, . . . , Vℓ are pairwise intersecting or share an edge. k + 1 vertices are needed to hit V1, . . . , Vℓ.

Cliques of size k : a bramble of order (k − 1) where Vi = {i}.
Trees : Two incident subtrees form a bramble of order 1.

a b c d e f g h 16/32

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Examples

A set of vertices V1, . . . , Vℓ is a bramble of order k if The sets V1, . . . , Vℓ are connected. The sets V1, . . . , Vℓ are pairwise intersecting or share an edge. k + 1 vertices are needed to hit V1, . . . , Vℓ.

Cliques of size k : a bramble of order (k − 1) where Vi = {i}.
Trees : Two incident subtrees form a bramble of order 1.
The d × d grid has a bramble of order d (Vi is the i-row

union the i-th column) :

Every pair row-column of the (d − 1) × (d − 1) grid.
The last row.
The last column minus the last vertex.

a b c d e f g h

a b c d e f g h i

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SLIDE 40

A new invariant : vinewidth

A tree T and a function f : T → 2V is a vine decomposition of G = (V , E) if :

For every v ∈ V , the set of nodes containing v in their bag is

a subtree Tv of T.

For every edge uv, Tu and Tv intersects or share an edge.

The width of a decomposition is the maximum size of a bag of the vine-decomposition. The vinewidth of G, is the minimum width of a tree- decomposition of G. Definition (vinewidth)

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SLIDE 41

A new invariant : vinewidth

A tree T and a function f : T → 2V is a vine decomposition of G = (V , E) if :

For every v ∈ V , the set of nodes containing v in their bag is

a subtree Tv of T.

For every edge uv, Tu and Tv intersects or share an edge.

The width of a decomposition is the maximum size of a bag of the vine-decomposition. The vinewidth of G, is the minimum width of a tree- decomposition of G. Definition (vinewidth)

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Examples

Kn has a vine decomposition of width ⌈n/2⌉.

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Examples

Kn has a vine decomposition of width ⌈n/2⌉.
Trees have vine decompositions of width 1.

a b c d e f g h a b c d e f g h

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Examples

Kn has a vine decomposition of width ⌈n/2⌉.
Trees have vine decompositions of width 1.
The d × d grid has a vine decomposition of width d.

a b c d e f g h a b c d e f g h a b c d e f g h i a, b c d, e f g, h i

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Dual notion : thicket

A set of vertices V1, . . . , Vℓ is a thicket of order k if

For every i, the set Vi is connected.
For every i = j, Vi and Vj intersect.
The minimum number of vertices intersecting all the sets

V1, . . . , Vℓ is k. Definition (thicket)

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Dual notion : thicket

A set of vertices V1, . . . , Vℓ is a thicket of order k if

For every i, the set Vi is connected.
For every i = j, Vi and Vj intersect.
The minimum number of vertices intersecting all the sets

V1, . . . , Vℓ is k. Definition (thicket) The vinewidth of the graph is equal to the maximum size of a thicket. Theorem (B., Li, Vetta)

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Dual notion : thicket

A set of vertices V1, . . . , Vℓ is a thicket of order k if

For every i, the set Vi is connected.
For every i = j, Vi and Vj intersect.
The minimum number of vertices intersecting all the sets

V1, . . . , Vℓ is k. Definition (thicket) The vinewidth of the graph is equal to the maximum size of a thicket. Theorem (B., Li, Vetta)

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Examples

A set of vertices V1, . . . , Vℓ is a thicket of order k if The sets V1, . . . , Vℓ are connected. The sets V1, . . . , Vℓ are pairwise intersecting. The minimum number of vertices intersecting all the sets V1, . . . , Vℓ is k.

Trees : Trees have the Helly property, thus the order of a

thicket is 1.

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SLIDE 49

Examples

A set of vertices V1, . . . , Vℓ is a thicket of order k if The sets V1, . . . , Vℓ are connected. The sets V1, . . . , Vℓ are pairwise intersecting. The minimum number of vertices intersecting all the sets V1, . . . , Vℓ is k.

Trees : Trees have the Helly property, thus the order of a

thicket is 1.

Cliques of size k :
Every subset of size more than k/2 is in the thicket.
Covering : ⌈k/2⌉.

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SLIDE 50

Examples

A set of vertices V1, . . . , Vℓ is a thicket of order k if The sets V1, . . . , Vℓ are connected. The sets V1, . . . , Vℓ are pairwise intersecting. The minimum number of vertices intersecting all the sets V1, . . . , Vℓ is k.

Trees : Trees have the Helly property, thus the order of a

thicket is 1.

Cliques of size k :
Every subset of size more than k/2 is in the thicket.
Covering : ⌈k/2⌉.
d × d grid.
Every pair row-column is in the thicket.
Covering : d.

a b c d e f g h i

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SLIDE 51

Link between treewidth and vinewidth

Every graph G satisfies tw(G) + 1 2 ≤ vw(G) ≤ tw(G) + 1 Moreover both inequalities are tight. Lemma

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SLIDE 52

Link between treewidth and vinewidth

Every graph G satisfies tw(G) + 1 2 ≤ vw(G) ≤ tw(G) + 1 Moreover both inequalities are tight. Lemma Proof sketch :

Any tree decomposition is a vine decomposition.

⇒ vw(G) ≤ tw(G) + 1.

We make the “union” of every pair of adjacent bags to be

sure that Tu and Tv intersect. ⇒ tw(G) ≤ 2 · vw(G) − 1.

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SLIDE 53

Main statement

τ = min

i∈I

xi s.t.

i:i∈S

xi ≥ v(S) ∀S ⊆ I ν = max

S:S⊆I

v(S) · yS s.t.

S⊆I:i∈S

yS ≤ 1 ∀i ∈ I

For every graph G, we have : vw(G) ≤∃ τ(G) ν(G) ≤∀ vw(G) Theorem (BLV’14) By ≤∀, we mean that every game G on interaction graph G satisfies this inequality. By ≤∃, we mean that there exists a game G on interaction graph G which satisfies this inequality.

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SLIDE 54

Proof of vw(G) ≤∃

τ(G) ν(G).

Take a thicket V1, . . . , Vℓ of order vw(G). We consider the following 0 − 1 game G where :

For every i, Vi is a coalition of value 1.
The other sets receive value 0.

Now, we have :

For every i, the set Vi is connected : G is a game on

interaction graph G.

All the sets of a thicket intersect : ν(G) = 1.
By definition of thicket : τ(G) = vw(G).

Thus τ(G)

ν(G) = vw(G)

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SLIDE 55

Proof of τ(G)

ν(G) ≤∀ vw(G).

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SLIDE 56

Proof of τ(G)

ν(G) ≤∀ vw(G).

Bottom-up from the leaves of a (rooted) vine decomposition.

If all the vertices of a coalition C are in the bag Bf of a leaf f ,

add vertices of Bf in the covering and C in the packing. Delete the coalitions containing one vertex of Bf .

Otherwise, delete all the leaves of T.

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SLIDE 57

Proof of τ(G)

ν(G) ≤∀ vw(G).

Bottom-up from the leaves of a (rooted) vine decomposition.

If all the vertices of a coalition C are in the bag Bf of a leaf f ,

add vertices of Bf in the covering and C in the packing. Delete the coalitions containing one vertex of Bf .

Otherwise, delete all the leaves of T.

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SLIDE 58

Proof of τ(G)

ν(G) ≤∀ vw(G).

Bottom-up from the leaves of a (rooted) vine decomposition.

If all the vertices of a coalition C are in the bag Bf of a leaf f ,

add vertices of Bf in the covering and C in the packing. Delete the coalitions containing one vertex of Bf .

Otherwise, delete all the leaves of T.

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SLIDE 59

Proof of τ(G)

ν(G) ≤∀ vw(G).

Bottom-up from the leaves of a (rooted) vine decomposition.

If all the vertices of a coalition C are in the bag Bf of a leaf f ,

add vertices of Bf in the covering and C in the packing. Delete the coalitions containing one vertex of Bf .

Otherwise, delete all the leaves of T.

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SLIDE 60

Proof of τ(G)

ν(G) ≤∀ vw(G).

Bottom-up from the leaves of a (rooted) vine decomposition.

If all the vertices of a coalition C are in the bag Bf of a leaf f ,

add vertices of Bf in the covering and C in the packing. Delete the coalitions containing one vertex of Bf .

Otherwise, delete all the leaves of T.
We have selected k disjoint coalitions : ν(G) ≥ k.
We have deleted at most vw(G) · k vertices.

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SLIDE 61

Other integrality gaps

Since the vinewidth correctly evaluates the packing-covering ra- tio, does it also correctly evaluates both integrality gaps ? Question Answer : YES (even if it is not tight) !

One integrality gap is linear in terms of the vinewidth.
One integrality gap is polynomial in terms of the vinewidth.

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SLIDE 62

Fractional packings and coverings

ν∗ = max

S:S⊆I

v(S) · yS s.t.

S⊆I:i∈S

yS ≤ 1 ∀i ∈ I yS ≥ 0 ∀S ⊆ I

The vector y can be seen as a weight function on the coalitions such that :

The total weight is

maximized.

For each vertex v, the sum
f the weights of the

coalitions containing v is at most 1.

τ ∗ = min

i∈I

xi s.t.

i:i∈S

xi ≥ v(S) ∀S ⊆ I xi ≥ ∀i ∈ I

The vector x can be seen as a weight function on the vertices such that :

The total weight is

minimized.

The weight of every

coalition S is at least v(S).

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SLIDE 63

Fractional packings and coverings

ν∗ = max

S:S⊆I

v(S) · yS s.t.

S⊆I:i∈S

yS ≤ 1 ∀i ∈ I yS ≥ 0 ∀S ⊆ I

The vector y can be seen as a weight function on the coalitions such that :

The total weight is

maximized.

For each vertex v, the sum
f the weights of the

coalitions containing v is at most 1.

τ ∗ = min

i∈I

xi s.t.

i:i∈S

xi ≥ v(S) ∀S ⊆ I xi ≥ ∀i ∈ I

The vector x can be seen as a weight function on the vertices such that :

The total weight is

minimized.

The weight of every

coalition S is at least v(S). By Strong Duality Theorem, we have τ ∗ = ν∗.

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SLIDE 64

Relative cost of stability

Reminder of the motivation : We want to bound the relative cost of stability τ ∗

ν .

Let G be a graph. There exists δ > 0 such that vw(G)δ ≤∃ ν∗(G) ν(G) ≤∀ vw(G) Theorem

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SLIDE 65

Relative cost of stability

Reminder of the motivation : We want to bound the relative cost of stability τ ∗

ν .

Let G be a graph. There exists δ > 0 such that vw(G)δ ≤∃ ν∗(G) ν(G) ≤∀ vw(G) Theorem Proof of ν∗(G)

ν(G) ≤∀ vw(G) :

Follows from the previous theorem since ν∗(G)

ν(G) ≤ τ(G) ν(G) ≤∀ vw(G).

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SLIDE 66

Relative cost of stability

Reminder of the motivation : We want to bound the relative cost of stability τ ∗

ν .

Let G be a graph. There exists δ > 0 such that vw(G)δ ≤∃ ν∗(G) ν(G) ≤∀ vw(G) Theorem Proof of ν∗(G)

ν(G) ≤∀ vw(G) :

Follows from the previous theorem since ν∗(G)

ν(G) ≤ τ(G) ν(G) ≤∀ vw(G).

Proof of vw(G)δ ≤∃

ν∗(G) ν(G) :

Prove that the gap is linear for grids.
Use grid minor theorem.

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SLIDE 67

Relative cost of stability on grids

There exist c′ and δ > 0 such that every graph G has a grid minor of size at least c′ · tw(G)δ. Theorem (Chekuri,Chuzhoy)

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SLIDE 68

Relative cost of stability on grids

There exist c′ and δ > 0 such that every graph G has a grid minor of size at least c′ · tw(G)δ. Theorem (Chekuri,Chuzhoy) Let Gd be the d × d grid (minor). We have : 1 2 · vw(Gd) ≤∃ ν∗(G) ν(G) Lemma

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SLIDE 69

Relative cost of stability on grids

There exist c′ and δ > 0 such that every graph G has a grid minor of size at least c′ · tw(G)δ. Theorem (Chekuri,Chuzhoy) Let Gd be the d × d grid (minor). We have : 1 2 · vw(Gd) ≤∃ ν∗(G) ν(G) Lemma Proof : Consider the game G where coalitions are Ri ∪ Ci for every i.

ν(G) = 1 since all the coalitions intersect.
ν∗(G) ≥ d

2 = vw(Gd) 2

since if we allocate weight 1

2 to each

coalition, the constraints are satisfied. ⇒ 1

2 · vw(G) ≤∃ ν∗(G) ν(G)

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SLIDE 70

Questions

vw(G)δ ≤∃ ν∗(G) ν(G) ≤∀ vw(G) Questions :

What is the best constant δ (we cannot beat 1

2, on cliques) ?

Which conditions ensure a linear bound ?
Does it exist a “nice” invariant catching the relative cost of

stability ?

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SLIDE 71

The other integrality gap

Let G be a graph. The following inequalities are satisfied : 1 4 · vw(G) ≤∃ τ(G) τ ∗(G) ≤∀ vw(G) Moreover, the right inequality is tight (in the limit). Theorem

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SLIDE 72

The other integrality gap

Let G be a graph. The following inequalities are satisfied : 1 4 · vw(G) ≤∃ τ(G) τ ∗(G) ≤∀ vw(G) Moreover, the right inequality is tight (in the limit). Theorem

In the LHS, vw(G) can be replaced by (tw(G) + 1) (which is

slightly better).

1

4 · tw(G) tight on cliques (and on grids ?).

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SLIDE 73

Proof of 1

4 · vw(G) ≤∃ τ(G) τ ∗(G) :

Let V1, . . . , Vℓ be a thicket of order k. Let Y = {y1, . . . , yk} be a hitting set of V1, . . . , Vℓ. Let G be the game such that S is a coalition iff :

S = ∪i∈J⊆{1,...,ℓ}Vi.
|S ∩ Y | > k/2.

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SLIDE 74

Proof of 1

4 · vw(G) ≤∃ τ(G) τ ∗(G) :

Let V1, . . . , Vℓ be a thicket of order k. Let Y = {y1, . . . , yk} be a hitting set of V1, . . . , Vℓ. Let G be the game such that S is a coalition iff :

S = ∪i∈J⊆{1,...,ℓ}Vi.
|S ∩ Y | > k/2.

Remark : For every yj there exists Vij such that Vij ∩ Y = {yj}.

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SLIDE 75

Proof of 1

4 · vw(G) ≤∃ τ(G) τ ∗(G) :

Let V1, . . . , Vℓ be a thicket of order k. Let Y = {y1, . . . , yk} be a hitting set of V1, . . . , Vℓ. Let G be the game such that S is a coalition iff :

S = ∪i∈J⊆{1,...,ℓ}Vi.
|S ∩ Y | > k/2.

Remark : For every yj there exists Vij such that Vij ∩ Y = {yj}. We have :

τ(G) ≥ k/2 since if Y ′ ⊆ Y satisfies |Y ′| > k/2 then

∪yj∈Y ′Vij is a coalition.

τ ∗(G) ≤ 2 since the weight function assigning weight
2/k to each vertex of Y ,
0 to the other vertices,

satisfies the constraints.

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SLIDE 76

Conclusion

Other applications of the vinewidth / thicket ?
Does it exist a “good” invariant which characterizes the

relative cost of stability ? The other integrality gap ?

Can we close the gap between lower and upper bounds for

grids ?

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SLIDE 77

Conclusion

Other applications of the vinewidth / thicket ?
Does it exist a “good” invariant which characterizes the

relative cost of stability ? The other integrality gap ?

Can we close the gap between lower and upper bounds for

grids ? Thanks for your attention !

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