Coalition games on interaction graphs Lets play with tree - - PowerPoint PPT Presentation

Coalition games

• n interaction graphs

Let’s play with tree decompositions Nicolas Bousquet

joint work with Zhentao Li and Adrian Vetta

S´ eminaire Complex Networks

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Context

• We want people to work together.

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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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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 the grand coalition (coalition of all agents).

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Coalition games

• A set I of n agents.
• A superadditive valuation function v : 2n → N. (the money

generated by the coalition S if agents of S decide to work on their

• wn project)

Coalition game superadditive : S ∩ T = ∅ ⇒ v(S ∪ T) ≥ v(S) + v(T).

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Coalition games

• A set I of n agents.
• A superadditive valuation function v : 2n → N. (the money

generated by the coalition S if agents of S decide to work on their

• wn project)

Coalition game superadditive : S ∩ T = ∅ ⇒ v(S ∪ T) ≥ v(S) + v(T). Distribute payoff to the agents in such a way, for every coalition S, the money distributed to agents of S is at least v(S). ⇒ No coalition wishes to leave the grand coalition. Goal

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Illustration

All the results of this talk are true for any superadditive valuation

• function. However, for simplicity, we will focus on simple games.

There exists a set X = {X1, . . . , Xm} of non empty subsets of I called minimal coalitions such that :

• v(Xi) for every i.
• The value of any set Y equals the maximum number of

pairwise disjoint elements of X in Y . Definition (simple games)

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Illustration

All the results of this talk are true for any superadditive valuation

• function. However, for simplicity, we will focus on simple games.

There exists a set X = {X1, . . . , Xm} of non empty subsets of I called minimal coalitions such that :

• v(Xi) for every i.
• The value of any set Y equals the maximum number of

pairwise disjoint elements of X in Y . Definition (simple games)

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Illustration

All the results of this talk are true for any superadditive valuation

• function. However, for simplicity, we will focus on simple games.

There exists a set X = {X1, . . . , Xm} of non empty subsets of I called minimal coalitions such that :

• v(Xi) for every i.
• The value of any set Y equals the maximum number of

pairwise disjoint elements of X in Y . Definition (simple games) ⇒ v(I) = 3.

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Computing v(I) via a Linear Program

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Computing v(I) via a Linear Program

Using a linear program called the (integral) Packing LP of the game G :

• We create an integral variable yS for each minimal coalition S.

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Computing v(I) via a Linear Program

Using a linear program called the (integral) Packing LP of the game G :

• We create an integral variable yS for each minimal coalition S.
• The goal consists in finding the maximum number of

coalitions. Pack(G) = max

• S:S⊆I

yS s.t. yS ∈ {0, 1} ∀S ⊆ I

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Computing v(I) via a Linear Program

Using a linear program called the (integral) Packing LP of the game G :

• We create an integral variable yS for each minimal coalition S.
• The goal consists in finding the maximum number of disjoint

coalitions. Pack(G) = max

• S:S⊆I

yS s.t.

• S⊆I:i∈S

yS ≤ 1 ∀i ∈ I yS ∈ {0, 1} ∀S ⊆ I

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Core

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

• i∈I xi = v(I)

The money we can distribute

xi ≥ 0 ∀i ∈ I

Non-negative salary

Definition (core)

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Core

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

• i∈I xi = v(I)

The money we can distribute

• i∈S xi ≥ v(S)

∀S ⊆ I

No coalition can benefit by deviating

xi ≥ 0 ∀i ∈ I

Non-negative salary

Definition (core)

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Core

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

• i∈I xi = v(I)

The money we can distribute

• i∈S xi ≥ v(S)

∀S ⊆ I

No coalition can benefit by deviating

xi ≥ 0 ∀i ∈ I

Non-negative salary

Definition (core)

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Core

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

• i∈I xi = v(I)

The money we can distribute

• i∈S xi ≥ v(S)

∀S ⊆ I

No coalition can benefit by deviating

xi ≥ 0 ∀i ∈ I

Non-negative salary

Definition (core)

• Problem : The core may be empty !
• Which conditions ensure that

the core is not empty ?

• Relax the definition of core.

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Relative cost of stability

The relative cost of stability evaluates how much money must be injected by an external authority to stabilize the system. ⇒ Our expenses minus our gains.

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Relative cost of stability

The relative cost of stability evaluates how much money must be injected by an external authority to stabilize the system. ⇒ Our expenses minus our gains. Our gains : v(I) = Pack(G).

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Relative cost of stability

The relative cost of stability evaluates how much money must be injected by an external authority to stabilize the system. ⇒ Our expenses minus our gains. Our gains : v(I) = Pack(G). Our expenses : Fractional covering. Cov(G)∗ = min

i∈I

xi

minimize the total cost

s.t.

• i:i∈S

xi ≥ v(S) ∀S ⊆ I

stability of each minimal coalition

xi ≥ 0

non negative salary

It represents the amount of money which has to be spent in order to stabilize the system.

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Relative cost of stability

The relative cost of stability evaluates how much money must be injected by an external authority to stabilize the system. ⇒ Our expenses minus our gains. Our gains : v(I) = Pack(G). Our expenses : Fractional covering. Cov(G)∗ = min

i∈I

xi

minimize the total cost

s.t.

• i:i∈S

xi ≥ v(S) ∀S ⊆ I

stability of each minimal coalition

xi ≥ 0

non negative salary

It represents the amount of money which has to be spent in order to stabilize the system. Stability of the notion : since Cov(G)∗ − Pack(G) is not “stable” (by disjoint copy for instance), we consider Cov(G)∗

Pack(G) instead.

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Relative Cost of Stability

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

Pack(G) .

Definition (relative cost of stability RCoS)

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Relative Cost of Stability

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

Pack(G) .

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

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Relative Cost of Stability

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

Pack(G) .

Definition (relative cost of stability RCoS) Game theoretical interpretation : The relative cost of stability represents the ratio between the minimum payment stabilizing the system and the total wealth the grand coalition can generate. By Strong Duality Theorem, we have : Pack(G) ≤ Pack∗(G) = Cov∗(G) ≤ Cov(G) Thus 1 ≤ Cov(G)∗ Pack(G) = Pack(G)∗ Pack(G) ≤ Cov(G) Pack(G)

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Integral covering and packing

Cov(G) = min

• i∈I

xi s.t.

• i:i∈S

xi ≥ v(S) ∀S ⊆ I xi ∈ {0, 1} ∀i ∈ I

• In other words the integral covering corresponds to the minimum number
• f agents intersecting all the minimal coalitions.

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Integral covering and packing

Cov(G) = min

• i∈I

xi s.t.

• i:i∈S

xi ≥ v(S) ∀S ⊆ I xi ∈ {0, 1} ∀i ∈ I

• In other words the integral covering corresponds to the minimum number
• f agents intersecting all the minimal coalitions.

Pack(G) = max

• S:S⊆I

yS s.t.

• S⊆I:i∈S

yS ≤ 1 ∀i ∈ I yS ∈ {0, 1} ∀S ⊆ I In other words the integral packing corresponds to the maximum number

• f disjoint minimal coalitions.

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Worst case

The Packing-Covering ratio

Cov(G) Pack(G) (and both integrality gaps)

can be arbitrarily large. Lemma

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Worst case

The Packing-Covering ratio

Cov(G) Pack(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 minimal coalition if and only if |S| = |I|/2.
• Maximum Packing : 1.

(Any pair of coalitions intersect).

• Minimum Covering : ≥ |I|

2 − 1.

Any smaller set of agents contain at least |I|/2 agents in their complement, and then a minimal coalition.

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Worst case

The Packing-Covering ratio

Cov(G) Pack(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 minimal coalition if and only if |S| = |I|/2.
• Maximum Packing : 1.

(Any pair of coalitions intersect).

• Minimum Covering : ≥ |I|

2 − 1.

Any smaller set of agents contain at least |I|/2 agents in their complement, and then a minimal coalition.

Which conditions imply that

Cov(G) Pack(G) is bounded ?

Question

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Interaction graph

Myerson proposed the following model 1 : the agents must be able to communicate if they want to form a viable coalition.

• 1. Conference structures and fair allocation rules, Myerson (IJGT 1980).

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Interaction graph

Myerson proposed the following model 1 : the agents must be able to communicate if they want to form a viable coalition. Let H be a graph where :

• Vertices = agents.
• Edges = ability to communicate.

The game G in on interaction graph H if minimal coalitions are connected subgraphs of H. Definition (interaction graph)

• 1. Conference structures and fair allocation rules, Myerson (IJGT 1980).

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Interaction graph

Myerson proposed the following model 1 : the agents must be able to communicate if they want to form a viable coalition. Let H be a graph where :

• Vertices = agents.
• Edges = ability to communicate.

The game G in on interaction graph H if minimal coalitions are connected subgraphs of H. Definition (interaction graph) Examples :

• H is a clique : any set of minimal coalitions is possible.
• H is a stable set : minimal coalitions have size one.
• 1. Conference structures and fair allocation rules, Myerson (IJGT 1980).

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Treewidth and coalition game

Let H be a graph. The Relative Cost of stability Cov∗(G)

Pack(G) of any

coalition game G on interaction graph H is at most tw(H) + 1. Moreover there exist graphs for which this bound is tight. Theorem (Meir et al.) 2

• 2. Bounding the cost of stability in games over interaction networks, Meir,

Zick, Elkind and Rosenschein (AAAI 2013).

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Treewidth and coalition game

Let H be a graph. The Relative Cost of stability Cov∗(G)

Pack(G) of any

coalition game G on interaction graph H is at most tw(H) + 1. Moreover there exist graphs for which this bound is tight. Theorem (Meir et al.) 2 Actually, they proved the following stronger statement : The following inequality holds : 4 Cov(G) Pack(G) ≤∀ tw(H) + 1 Theorem (Meir et al.)

Cov(G) = min

• i∈I

xi s.t. ∀S ⊆ I

• i:i∈S

xi ≥ v(S) xi ∈ N

• 2. Bounding the cost of stability in games over interaction networks, Meir,

Zick, Elkind and Rosenschein (AAAI 2013).

• 4. ≤∀ means that every G on interaction graph H satisfies this inequality.

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Our results

We introduce a new invariant vw(H) that completely characterizes the packing-covering ratio, i.e. for every graph H : 3 vw(H) ≤∃

Cov(G) Pack(G) ≤∀ vw(H)

Informal Result 1

• 3. ≤∃

means that there exists a game G on interaction graph H which satisfies this inequality. ≤∀ means that every game G on interaction graph H satisfies this inequality.

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Our results

We introduce a new invariant vw(H) that completely characterizes the packing-covering ratio, i.e. for every graph H : 3 vw(H) ≤∃

Cov(G) Pack(G) ≤∀ vw(H)

Informal Result 1 There exists δ > 0 such that for every graph H, we have vw(H)δ ≤∃ RCoS(G) = Cov ∗(G) Pack(G) ≤∀ vw(H) Informal Result 2

• 3. ≤∃

means that there exists a game G on interaction graph H which satisfies this inequality. ≤∀ means that every game G on interaction graph H satisfies this inequality.

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Our results

We introduce a new invariant vw(H) that completely characterizes the packing-covering ratio, i.e. for every graph H : 3 vw(H) ≤∃

Cov(G) Pack(G) ≤∀ vw(H)

Informal Result 1 There exists δ > 0 such that for every graph H, we have vw(H)δ ≤∃ RCoS(G) = Cov ∗(G) Pack(G) ≤∀ vw(H) Informal Result 2 There exists a constant c such that c · vw(H) ≤∃ Cov(G) Cov ∗(G) ≤∀ vw(H) Informal Result 3

• 3. ≤∃

means that there exists a game G on interaction graph H which satisfies this inequality. ≤∀ means that every game G on interaction graph H satisfies this inequality.

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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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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 17/30

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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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 vine- decomposition of G. Definition (vinewidth)

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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 vine- 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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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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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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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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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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Main statement

Cov = min

• i∈I

xi s.t.

• i:i∈S

xi ≥ v(S) ∀S ⊆ I Pack = max

• S:S⊆I

yS s.t.

• S⊆I:i∈S

yS ≤ 1 ∀i ∈ I

For every graph G, we have : vw(G) ≤∃ Cov(G) Pack(G) ≤∀ vw(G) Theorem Reminder : 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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Proof of vw(G) ≤∃

Cov(G) Pack(G).

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

• Minimal coalitions are the coalitions V1, . . . , Vℓ.

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Proof of vw(G) ≤∃

Cov(G) Pack(G).

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

• Minimal coalitions are the coalitions V1, . . . , Vℓ.

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 : Pack(G) = 1.
• By definition of thicket : Cov(G) = vw(G).

Thus

Cov(G) Pack(G) = vw(G)

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Proof of

Cov(G) Pack(G) ≤∀ vw(G).

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Proof of

Cov(G) Pack(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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Proof of

Cov(G) Pack(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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Proof of

Cov(G) Pack(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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Proof of

Cov(G) Pack(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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Proof of

Cov(G) Pack(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 : Pack(G) ≥ k.
• We have deleted at most vw(G) · k vertices.

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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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Fractional packings and coverings

Pack∗ = max

• S:S⊆I

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.

Cov ∗ = 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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Fractional packings and coverings

Pack∗ = max

• S:S⊆I

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.

Cov ∗ = 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 Cov∗ = Pack∗.

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Relative cost of stability

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

Pack .

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

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Relative cost of stability

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

Pack .

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

Pack(G) ≤∀ vw(G) :

Follows from the previous theorem since

Pack∗(G) Pack(G) ≤ Cov(G) Pack(G) ≤∀ vw(G).

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Relative cost of stability

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

Pack .

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

Pack(G) ≤∀ vw(G) :

Follows from the previous theorem since

Pack∗(G) Pack(G) ≤ Cov(G) Pack(G) ≤∀ vw(G).

Proof of vw(G)δ ≤∃

Pack∗(G) Pack(G) :

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

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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 ’14)

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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 ’14) Let Gd be the d × d grid (minor). We have : 1 2 · vw(Gd) ≤∃ Pack∗(G) Pack(G) Lemma

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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 ’14) Let Gd be the d × d grid (minor). We have : 1 2 · vw(Gd) ≤∃ Pack∗(G) Pack(G) Lemma

a b c d e f g h i

Proof : Game G with minimal coalitions are Ri ∪ Ci. Pack(G) = 1 (coalitions pairwise intersect). Pack∗(G) ≥ d

2 = vw(Gd) 2 (allocate weight 1

2 to each

coalition).

⇒ 1

2 · vw(G) ≤∃ Pack∗(G) Pack(G)

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Questions

vw(G)δ ≤∃ Pack∗(G) Pack(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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Questions

vw(G)δ ≤∃ Pack∗(G) Pack(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 ? Further work :

• Other applications of the vinewidth / thicket duality ?
• What is the best coefficient δ ? (δ cannot be improved beyond

1 2)

• Equivalence between the coefficient of the RCoS and of the

grid minor theorem ?

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Questions

vw(G)δ ≤∃ Pack∗(G) Pack(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 ? Further work :

• Other applications of the vinewidth / thicket duality ?
• What is the best coefficient δ ? (δ cannot be improved beyond

1 2)

• Equivalence between the coefficient of the RCoS and of the

grid minor theorem ? Thanks for your attention !

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