Coalition games on interaction graphs

Coalition games on interaction graphs

Nicolas Bousquet

Zhentao Li

Adrian Vetta LIRIS,

Lyon

DIENS, Paris

McGill University,

Canada

Coalition games v

I : agents

Input:

v

v(I)

v(S)

v(T)

v : valuations

Coalition games

I : agents

Input:

v

Goal: Distribute total wealth to agents to avoid

blocking coalitions.

v(I)

v(S)

v(T)

v : valuations

Coalition games

I : agents

Input:

Goal: Distribute to avoid

Coalition games

v(I)

v

v(T)=8

2 2 1

0 3

2 1

1 1

v(I)=13

v(T) > x(T)

T is blocking

Goal: Distribute to avoid

Coalition games

v(I)

v

v(T)=8

2 2 1

0 3

2 1

1 1

v(I)=13

v(T) > x(T)

for disjoint S, T T is blocking

Goal: Distribute to avoid

Coalition games

v(I)

v

v(T)=8

2 2 1

0 3

2 1

1 1

v(I)=13

v(T) > x(T)

Goal: Find x s.t.

LP formulation

2 2 1

0 3

2 1

1 1

v(I)

where

core : all x satisfying this Goal: Find x s.t.

LP formulation

2 2 1

0 3

2 1

1 1

v(I)

where

Communication

Communication

connected

= possible disconnected

= impossible

Communication

v(S)=0

connected

= possible disconnected

= impossible

Communication

Example graphs

Same as unrestricted

Only coalitions of size 1

Example graphs

Same as unrestricted

Goal: Find x s.t.

Least core

core : all x satisfying this

Goal: Find x s.t.

Least core

-core : all x satisfying this

: minimum with non-empty -core least-core : -core

Goal: Find x s.t.

Least core

-core : all x satisfying this

= Relative cost of stability : minimum with

non-empty -core least-core : -core

Goal: Find x s.t.

Least core

-core : all x satisfying this

Packing-covering LP

Covering LP

s.t.

s.t.

Packing LP

Packing-covering LP

Covering LP

s.t.

s.t.

Packing LP

Packing-covering LP

Covering LP

s.t.

optimum

integral opt is

s.t.

Packing LP

Packing-covering LP

Covering LP

s.t.

Main question

Question: Given a graph, what is the worst possible ratio

over all valuations?

Main question

Question: Given a graph, what is the worst possible ratio

over all valuations?

Previous bound: treewidth

- this inequality holds for all games - treewidth of

Meir et al.

Treewidth

A B

C

D E F G

H

Treewidth

A B

C

D E F

G

H

Tree decomposion of

Treewidth

A B

C

D E F

G

H

Tree decomposion of

A B C

C C

B B

B

D E

E E

E F G G

G H

Treewidth

A B

C

D E F

G

H

Tree decomposion of

A B C

C C

B B

B

D E

E E

E F G G

G H

- subtree of nodes containing

Treewidth

A B

C

D E F

G

H

Tree decomposion of

B C

C C

B B

B

- subtree of nodes containing adjacent

vertices intersecting subtrees

Treewidth

A B

C

D E F

G

H

Tree decomposion of

A B C

C C

B B

B

D E

E E

E F G G

G H

- subtree of nodes containing of width 3-1 = 2 adjacent

vertices intersecting subtrees

- vinewidth of

Vine decomposion of of width 3

Vinewidth

B A

C DFE

A B

C D E

F

adjacent vertices

intersecting or adjacent subtrees

Our new bounds

- this inequality holds for some game - this inequality holds for all games

- vinewidth of

Upper bound

Proof idea:

Upper bound

Proof idea:

1 5

Upper bound

Proof idea:

1 5

Upper bound

Proof idea:

root root

1 5

Upper bound

Proof idea:

1

Find min payment to make x feasible.1

Pay that to all vertices in that node.

root root

1 5

Upper bound

Proof idea:

1

Find min payment to make x feasible.1

Pay that to all vertices in that node.

root root

1 5

Upper bound

Proof idea:

1

Find min payment to make x feasible.1

Pay that to all vertices in that node.

root root

1 5

Upper bound

Proof idea:

1

Find min payment to make x feasible.1

Pay that to all vertices in that node.

3 3 root root

Lower bound

Duality theorem:

vinewidth = maximum thicket order

Lower bound

thicket: pairwise intersecting connected subgraphs

order: smallest hitting set

Duality theorem:

vinewidth = maximum thicket order

Lower bound

thicket: pairwise intersecting connected subgraphs

order: smallest hitting set

Duality theorem:

vinewidth = maximum thicket order

Lower bound

Lower bound proof

thicket: pairwise intersecting connected subgraphs

order: smallest hitting set

Lower bound proof

thicket: pairwise intersecting connected subgraphs

order: smallest hitting set

Lower bound proof

So

thicket: pairwise intersecting connected subgraphs

order: smallest hitting set

Lower bound proof

And So

thicket: pairwise intersecting connected subgraphs

order: smallest hitting set

Lower bound proof

And So

thicket: pairwise intersecting connected subgraphs

order: smallest hitting set

- this inequality holds for some game - this inequality holds for all games

Our other bounds

Packing gap

Packing gap

Find a grid minor in G of size polynomial in tw(G)

Chekuri Chuzhoy

Packing gap

Find a grid minor in G of size polynomial in tw(G)

Chekuri Chuzhoy

Prove

Packing gap

Find a grid minor in G of size polynomial in tw(G)

Chekuri Chuzhoy

Prove

Build a game using this thicket in just the minor.

Covering gap

Covering gap

Take the best thicket H and a best hitting set X for it.

Covering gap

Take the best thicket H and a best hitting set X for it.

v(S) = 1 if S is union of elements of H and intersects > half of X

Covering gap

Take the best thicket H and a best hitting set X for it.

v(S) = 1 if S is union of elements of H and intersects > half of X

Assign 2/|Y| to all X

Covering gap

Take the best thicket H and a best hitting set X for it.

v(S) = 1 if S is union of elements of H and intersects > half of X

Assign 2/|Y| to all X

Thank you