Covering Games:
Approximation through Non-Cooperation
Martin Gairing
Warwick 2010
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
1
Introduction
Covering Games
A general covering problem
Given
I a universe E of elements
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
2
Introduction
Covering Games
A general covering problem
Given
I a universe E of elements
I
5
2
20
12
12
10
6
30
1
a weight function w : E 7→ N
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
2
Introduction
Covering Games
A general covering problem
Given
I a universe E of elements
I
I
a weight function w : E 7→ N
n collections of subsets of E
I
Si ⊂ 2E for each i ∈ [n]
Covering Games: Approximation through Non-Cooperation
5
2
20
12
12
10
6
30
1
Martin Gairing
·
2
Introduction
Covering Games
A general covering problem
Given
I a universe E of elements
I
I
a weight function w : E 7→ N
n collections of subsets of E
I
Si ⊂ 2E for each i ∈ [n]
Task
I choose n subsets (si )i∈[n] , s.t.
I
I
5
2
20
12
12
10
6
30
1
sSi ∈ Si
i∈[n] si has maximum total
weight
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
2
Introduction
Covering Games
Covering Games
I
n players
Covering Games: Approximation through Non-Cooperation
5
2
20
12
12
10
6
30
1
Martin Gairing
·
3
Introduction
Covering Games
Covering Games
I
n players
I
player i ∈ [n] chooses si ∈ Si
Covering Games: Approximation through Non-Cooperation
5
2
20
12
12
10
6
30
1
Martin Gairing
·
3
Introduction
Covering Games
Covering Games
I
n players
I
player i ∈ [n] chooses si ∈ Si
I
for covering an element, pay players
according to utility sharing function
I
5
2
20
12
12
10
6
30
1
f : [n] 7→ [0, 1]
f (1) = 1
Covering Games: Approximation through Non-Cooperation
f (2) =
1
2
f (3) =
Martin Gairing
·
1
3
3
Introduction
Covering Games
Covering Games
I
n players
I
player i ∈ [n] chooses si ∈ Si
I
for covering an element, pay players
according to utility sharing function
I
I
5
2
20
12
12
10
6
30
1
f : [n] 7→ [0, 1]
natural assumptions on f
I
I
non-increasing
no-overpay (j · f (j) ≤ 1)
f (1) = 1
Covering Games: Approximation through Non-Cooperation
f (2) =
1
2
f (3) =
Martin Gairing
·
1
3
3
Introduction
Covering Games
Covering Games
I
n players
I
player i ∈ [n] chooses si ∈ Si
I
for covering an element, pay players
according to utility sharing function
I
I
I
I
2
20
12
12
10
6
30
1
f : [n] 7→ [0, 1]
natural assumptions on f
I
I
5
non-increasing
no-overpay (j · f (j) ≤ 1)
Load on e ∈ E:
δe (s) = |{i ∈ [n] : e ∈ si }|
Utility of player i ∈ [n]:
X
f (δe (s)) · we
ui (s) =
e∈si
Covering Games: Approximation through Non-Cooperation
f (1) = 1
f (2) =
I
u1 (s) = 37
I
u2 (s) = 13
I
u3 (s) = 11
1
2
f (3) =
Martin Gairing
·
1
3
3
Introduction
Covering Games
Covering Games
I
n players
I
player i ∈ [n] chooses si ∈ Si
I
for covering an element, pay players
according to utility sharing function
I
I
I
I
2
20
12
12
10
6
30
1
f : [n] 7→ [0, 1]
natural assumptions on f
I
I
5
non-increasing
no-overpay (j · f (j) ≤ 1)
Load on e ∈ E:
δe (s) = |{i ∈ [n] : e ∈ si }|
Utility of player i ∈ [n]:
X
f (δe (s)) · we
ui (s) =
e∈si
Covering Games: Approximation through Non-Cooperation
f (1) = 1
f (2) =
I
u1 (s) = 31
I
u2 (s) = 7
I
u3 (s) = 6
1
3
f (3) =
1
6
:-(
Martin Gairing
·
3
Introduction
Covering Games
Covering Games
I
n players
I
player i ∈ [n] chooses si ∈ Si
I
for covering an element, pay players
according to utility sharing function
I
I
I
I
2
20
12
12
10
6
30
1
f : [n] 7→ [0, 1]
natural assumptions on f
I
I
5
non-increasing
no-overpay (j · f (j) ≤ 1)
Load on e ∈ E:
δe (s) = |{i ∈ [n] : e ∈ si }|
Utility of player i ∈ [n]:
X
f (δe (s)) · we
ui (s) =
e∈si
Covering Games: Approximation through Non-Cooperation
f (1) = 1
f (2) =
I
u1 (s) = 24
I
u2 (s) = 8
I
u3 (s) = 11
1
3
f (3) =
1
6
:-(
Martin Gairing
·
3
Introduction
Covering Games
Covering Games
I
n players
I
player i ∈ [n] chooses si ∈ Si
I
for covering an element, pay players
according to utility sharing function
I
I
I
I
2
20
12
12
10
6
30
1
f : [n] 7→ [0, 1]
natural assumptions on f
I
I
5
non-increasing
no-overpay (j · f (j) ≤ 1)
Load on e ∈ E:
δe (s) = |{i ∈ [n] : e ∈ si }|
Utility of player i ∈ [n]:
X
f (δe (s)) · we
ui (s) =
e∈si
Covering Games: Approximation through Non-Cooperation
f (1) = 1
f (2) =
I
u1 (s) = 30
I
u2 (s) = 24
I
u3 (s) = 31
1
3
f (3) =
Martin Gairing
·
1
6
3
Introduction
Covering Games
Special Cases
[ N EMHAUSER , W OLSEY, F ISHER, ’78]
MAX-k-Cover
Si = Sj for all players i, j ∈ [n]
Greedy ⇒ (1 − 1e ) − approx.
[ F EIGE, ’98]
better ⇒ NP ⊆ TIME(nO(log log n) )
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
4
Introduction
Covering Games
Special Cases
[ N EMHAUSER , W OLSEY, F ISHER, ’78]
MAX-k-Cover
Si = Sj for all players i, j ∈ [n]
SAT-Games
|Si | ≤ 2 for each player i ∈ [n]
Greedy ⇒ (1 − 1e ) − approx.
[ F EIGE, ’98]
better ⇒ NP ⊆ TIME(nO(log log n) )
[ G IANNAKOS ET AL ., ’07]
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
4
Introduction
Covering Games
Special Cases
[ N EMHAUSER , W OLSEY, F ISHER, ’78]
MAX-k-Cover
Si = Sj for all players i, j ∈ [n]
SAT-Games
|Si | ≤ 2 for each player i ∈ [n]
Market-Sharing Games
5
3
Greedy ⇒ (1 − 1e ) − approx.
[ F EIGE, ’98]
better ⇒ NP ⊆ TIME(nO(log log n) )
[ G IANNAKOS ET AL ., ’07]
[ G OEMANS , M IRROKNI , T HOTTAN, ’04]
4
markets
1
2
3
4
2
2
players
3
4
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
4
Introduction
Covering Games
Nash Equilibrium
Nash Equilibrium
The (pure) strategy profile s is a pure Nash equilibrium if and only if all
players i ∈ [n] are satisfied, that is,
ui (s) ≥ ui (s−i , si0 )
for all i ∈ [n] and si0 ∈ Si .
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
5
Introduction
Covering Games
Nash Equilibrium
Nash Equilibrium
The (pure) strategy profile s is a pure Nash equilibrium if and only if all
players i ∈ [n] are satisfied, that is,
ui (s) ≥ ui (s−i , si0 )
for all i ∈ [n] and si0 ∈ Si .
Proposition
[ R OSENTHAL, 1973]
Every covering game admits a pure Nash equilibrium.
Rosenthals potential function:
Φ(s) =
e (s)
X δX
fe (i)
e∈E i=1
If a single player increases her payoff by ∆ then also the potential
increases by ∆.
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
5
Covering Games
Price of Anarchy
Price of Anarchy
I
W (s) . . . total weight of elements covered in s
I
f . . . utility sharing function.
Price of Anarchy
PoAf =
inf
Γ∈G,
s is NE in Γ
W (s)
OPT
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
6
Covering Games
Price of Anarchy
Price of Anarchy
I
W (s) . . . total weight of elements covered in s
I
f . . . utility sharing function.
Price of Anarchy
PoAf =
inf
Γ∈G,
s is NE in Γ
W (s)
OPT
Main task
Construct utility sharing function that maximizes PoAf .
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
6
Covering Games
Price of Anarchy
Price of Anarchy
I
W (s) . . . total weight of elements covered in s
I
f . . . utility sharing function.
Price of Anarchy
PoAf =
inf
Γ∈G,
s is NE in Γ
W (s)
OPT
Main task
Construct utility sharing function that maximizes PoAf .
I
Coordination Mechanism
[ C HRISTODOULOU, KOUTSOUPIAS , N ANAVATI, ’04]
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
6
Covering Games
Upper Bound
What to hope for?
Example: k=4
I
I
node ↔ element (we = 1)
edge ↔ player
I
I
I
|S1 | = 1
|Si | = 2 for i ≥ 2
k + 1 levels
I
I
root: k − 1 children
level j node: k − j children
f : [n] 7→ R depends only on the number of players choosing an element.
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
7
Covering Games
Upper Bound
What to hope for?
Example: k=4
I
I
node ↔ element (we = 1)
edge ↔ player
I
I
I
|S1 | = 1
|Si | = 2 for i ≥ 2
k + 1 levels
I
I
root: k − 1 children
level j node: k − j children
f : [n] 7→ R depends only on the number of players choosing an element.
Optimum s∗
W (s∗ ) = 1 +
Covering Games: Approximation through Non-Cooperation
Pk
j=1 (k
− 1) ·
(k−1)!
(k−j)!
Martin Gairing
·
7
Covering Games
Upper Bound
What to hope for?
Example: k=4
I
I
node ↔ element (we = 1)
edge ↔ player
I
I
I
|S1 | = 1
|Si | = 2 for i ≥ 2
k + 1 levels
I
I
root: k − 1 children
level j node: k − j children
f : [n] 7→ R depends only on the number of players choosing an element.
Nash Equilibrium s
Pk−1
W (s) = 1 + j=1 (k − 1) ·
Optimum s∗
(k−1)!
(k −j)!
W (s∗ ) = 1 +
Covering Games: Approximation through Non-Cooperation
Pk
j=1 (k
− 1) ·
(k−1)!
(k−j)!
Martin Gairing
·
7
Covering Games
Upper Bound
What to hope for?
Example: k=4
I
I
node ↔ element (we = 1)
edge ↔ player
I
I
I
|S1 | = 1
|Si | = 2 for i ≥ 2
k + 1 levels
I
I
root: k − 1 children
level j node: k − j children
f : [n] 7→ R depends only on the number of players choosing an element.
Nash Equilibrium s
Pk−1
W (s) = 1 + j=1 (k − 1) ·
Optimum s∗
(k−1)!
(k −j)!
W (s∗ ) = 1 +
Pk
j=1 (k
− 1) ·
(k−1)!
(k−j)!
Theorem
PoAf (k) ≤ 1 −
1
1
(k −1)(k−1)!
Covering Games: Approximation through Non-Cooperation
+
Pk−1
1
j=0 j!
.
Martin Gairing
·
7
Covering Games
General Lower Bound
What is known?
A simple example shows:
I
If f is defined by f (j) =
⇒ PoAf ≤ 12
1
j
for all j ∈ N
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
8
Covering Games
General Lower Bound
What is known?
A simple example shows:
I
If f is defined by f (j) =
⇒ PoAf ≤ 12
1
j
for all j ∈ N
Consider utility sharing function which is
I
non-increasing,
I
j · f (j) ≤ 1 (no-overpay), and
I
f (1) = 1
Then the covering game is also a valid utility game.
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
8
Covering Games
General Lower Bound
What is known?
A simple example shows:
I
If f is defined by f (j) =
⇒ PoAf ≤ 12
1
j
for all j ∈ N
Consider utility sharing function which is
I
non-increasing,
I
j · f (j) ≤ 1 (no-overpay), and
I
f (1) = 1
Then the covering game is also a valid utility game.
Theorem
[ V ETTA, ’02]
PoAf ≥
1
2
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
8
Covering Games
General Lower Bound
General Lower Bound on PoA:
I
Given utility sharing function f
I
Define χ = χ(f ) as the smallest number, such that ∀j ∈ N:
j · f (j) − f (j + 1) ≤ χ · f (1)
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
9
Covering Games
General Lower Bound
General Lower Bound on PoA:
I
Given utility sharing function f
I
Define χ = χ(f ) as the smallest number, such that ∀j ∈ N:
j · f (j) − f (j + 1) ≤ χ · f (1)
Theorem
PoAf ≥
1
χ+1
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
9
Covering Games
General Lower Bound
General Lower Bound on PoA:
I
Given utility sharing function f
I
Define χ = χ(f ) as the smallest number, such that ∀j ∈ N:
j · f (j) − f (j + 1) ≤ χ · f (1)
Theorem
PoAf ≥
1
χ+1
Remarks
I Construct f such that χ is minimized.
Covering Games: Approximation through Non-Cooperation
Martin Gairing
·
9
Covering Games
Optimum Lower Bound
Construct f that minimizes χ
Task
min χ s.t.
I
i · f (i) − f (i + 1) ≤ χ · f (1)
I
(k − 1) · f (k) ≤ χ · f (1)
for all i ∈ [k − 1]
Covering Games: Approximation through Non-Cooperation
Martin Gairing
· 10
Covering Games
Optimum Lower Bound
Construct f that minimizes χ
Task
min χ s.t.
I
i · f (i) − f (i + 1) ≤ χ · f (1)
I
(k − 1) · f (k) ≤ χ · f (1)
0 1−χ
B −χ
B
B
.
B
.
B
.
B
B −χ
B
B
.
B
.
B
B
.
B
B −χ
@
−χ
−χ
−1
2
0
−1
.
0
for all i ∈ [k − 1]
.
.
...
0
0
.
.
i
0
0
...
...
...
0
0
0
···
···
0
···
.
−1
.
0
0
0
0
0
.
0
0
0
.
.
.
.
k −2
0
0
Covering Games: Approximation through Non-Cooperation
−1
k −1
0
0
0
.
.
.
0
.
.
.
0
−1
k −1
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
Martin Gairing
· 10
Covering Games
Optimum Lower Bound
Construct f that minimizes χ
Task
min χ s.t.
I
i · f (i) − f (i + 1) ≤ χ · f (1)
I
(k − 1) · f (k) ≤ χ · f (1)
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1−χ
−χ
.
.
.
−χ
.
.
.
−χ
−χ[1 + (k − 1)]
−χ
−1
2
0
−1
.
0
for all i ∈ [k − 1]
.
.
...
0
0
.
.
i
0
0
...
...
...
0
0
0
···
···
0
···
.
−1
.
0
0
0
0
0
.
0
0
0
.
.
.
.
k −2
0
0
Covering Games: Approximation through Non-Cooperation
−1
(k − 1)2
0
0
0
.
.
.
0
.
.
.
0
0
k −1
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
Martin Gairing
· 10
Covering Games
Optimum Lower Bound
Construct f that minimizes χ
Task
min χ s.t.
I
i · f (i) − f (i + 1) ≤ χ · f (1)
I
(k − 1) · f (k) ≤ χ · f (1)
1−χ
−χ
B
B
B
.
B
.
B
.
B
B
−χ
B
B
.
B
.
B
.
B
h
i
B
P
B −χ 1 + (k − 1) 2 (k −1)!
B
j=1 (k−j)!
@
−χ[1 + (k − 1)]
−χ
0
−1
2
for all i ∈ [k − 1]
0
−1
.
0
.
.
...
0
0
.
.
i
0
0
0
0
···
···
0
···
.
−1
.
.
.
.
.
.
0
0
.
.
.
0
.
.
.
0
...
0
0
(k − 1)2 (k − 2)
0
0
0
0
...
...
0
0
0
0
0
0
(k − 1)2
0
0
k −1
Covering Games: Approximation through Non-Cooperation
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
Martin Gairing
· 10
Covering Games
Optimum Lower Bound
Construct f that minimizes χ
Task
min χ s.t.
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
I
i · f (i) − f (i + 1) ≤ χ · f (1)
I
(k − 1) · f (k) ≤ χ · f (1)
h
Pk −1 (k −1)! i
(k − 1)(k − 1)! − χ 1 + (k − 1) j=1
(k −j)!
.
.
.
h
P −i (k−1)! i
−χ 1 + (k − 1) kj=1
(k −j)!
.
.
.
−χ[1 + (k − 1)]
−χ
for all i ∈ [k − 1]
0
0
.
0
0
.
..
...
0
(k −1)!
(k − 1) (i−1)!
...
...
Covering Games: Approximation through Non-Cooperation
···
..
0
.
0
0
0
0
0
..
0
0
0
.
···
(k − 1)
0
1
.
.
.
C
C
C
C
C
C
C
C
C
C
C
C
C
A
0
..
0
0
0
2
.
.
.
0
k −1
Martin Gairing
· 10
Covering Games
Optimum Lower Bound
Construct f that minimizes χ
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
h
i
P
(k −1)!
(k − 1)(k − 1)! − χ 1 + (k − 1) k−1
j=1 (k −j)!
.
.
.
i
h
P
(k −1)!
−χ 1 + (k − 1) k−i
j=1 (k −j)!
.
.
.
−χ[1 + (k − 1)]
−χ
0
0
.
0
.
0
.
.
...
.
0
.
(k −1)!
(k − 1) (i−1)!
0
.
0
0
...
...
0
0
···
0
.
0
0
···
0
.
.
.
0
0
1
.
.
.
C
C
C
C
C
C
C
C
C
C
C
C
C
A
0
.
(k − 1)
0
0
2
.
.
.
0
k −1
k is known
I
χ=
1
P −1 1
1
+ kj=1
(k−1)(k −1)!
j!
Covering Games: Approximation through Non-Cooperation
Martin Gairing
· 11
Covering Games
Optimum Lower Bound
Construct f that minimizes χ
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
h
i
P
(k −1)!
(k − 1)(k − 1)! − χ 1 + (k − 1) k−1
j=1 (k −j)!
.
.
.
i
h
P
(k −1)!
−χ 1 + (k − 1) k−i
j=1 (k −j)!
.
.
.
−χ[1 + (k − 1)]
−χ
0
0
.
0
.
0
.
.
...
.
0
.
(k −1)!
(k − 1) (i−1)!
0
.
0
0
...
...
0
0
···
0
.
0
0
···
0
.
.
.
0
0
1
.
.
.
C
C
C
C
C
C
C
C
C
C
C
C
C
A
0
.
(k − 1)
0
0
2
.
.
.
0
k −1
k is known
1
I
χ=
I
Utility sharing function: P
P −1 1
1
+ kj=1
(k−1)(k −1)!
j!
1
−1)!
f (i) = (i − 1)! (k −1)(k
1
(k −1)(k −1)!
+
+
k−1 1
j=i j!
1
j=1 j!
Pk−1
Covering Games: Approximation through Non-Cooperation
Martin Gairing
· 11
Covering Games
Optimum Lower Bound
Construct f that minimizes χ
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
h
i
P
(k −1)!
(k − 1)(k − 1)! − χ 1 + (k − 1) k−1
j=1 (k −j)!
.
.
.
i
h
P
(k −1)!
−χ 1 + (k − 1) k−i
j=1 (k −j)!
0
0
.
0
.
.
.
−χ[1 + (k − 1)]
−χ
.
0
.
.
...
.
0
.
(k −1)!
(k − 1) (i−1)!
0
.
0
0
...
...
0
0
···
0
.
0
0
···
0
.
.
.
0
0
1
.
.
.
C
C
C
C
C
C
C
C
C
C
C
C
C
A
0
.
(k − 1)
0
0
2
.
.
.
0
k −1
k is known
1
I
χ=
I
Utility sharing function: P
P −1 1
1
+ kj=1
(k−1)(k −1)!
j!
1
−1)!
f (i) = (i − 1)! (k −1)(k
1
(k −1)(k −1)!
I
PoAf ≥ 1 −
+
+
k−1 1
j=i j!
1
j=1 j!
Pk−1
1
P −1
1
+ kj=0
(k−1)(k−1)!
1
j!
Covering Games: Approximation through Non-Cooperation
Martin Gairing
· 11
Covering Games
Optimum Lower Bound
Construct f that minimizes χ
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
h
i
P
(k −1)!
(k − 1)(k − 1)! − χ 1 + (k − 1) k−1
j=1 (k −j)!
.
.
.
i
h
P
(k −1)!
−χ 1 + (k − 1) k−i
j=1 (k −j)!
.
.
.
−χ[1 + (k − 1)]
−χ
0
.
0
0
0
.
.
.
0
.
(k −1)!
(k − 1) (i−1)!
...
0
0
...
...
···
0
0
.
···
0
.
.
.
0
0
(k − 1)
0
1
.
.
.
C
C
C
C
C
C
C
C
C
C
C
C
C
A
0
.
0
0
0
2
.
.
.
0
k −1
k is unknown (k → ∞)
1
I
χ=
I
Utility sharing function: P
P −1 1
1
+ kj=1
(k−1)(k −1)!
j!
f (i) = (i − 1)!
PoAf ≥ 1 −
.
0
.
k is known
I
0
1
+ k−1
j=i
(k −1)(k −1)!
P
1
+ k−1
j=1
(k −1)(k −1)!
1
P −1 1
1
+ kj=0
(k−1)(k−1)!
j!
1
e−1
I
χ=
I
Utility sharing function:
P
1
j!
1
j!
f (i) = (i − 1)!
I
PoAf ≥ 1 −
Covering Games: Approximation through Non-Cooperation
e−
i−1 1
j=0 j!
e−1
1
e
Martin Gairing
· 11
Covering Games
Distributed (Local Search) Approximation Algorithm
Distributed Approximation Algorithm
Task
I
Turn this into (1 − 1e )-approximation algorithm.
I
I
I
Start with arbitrary strategy profile.
Let players unilaterally improve. (selfish steps)
Use Rosenthals potential function to bound running time.
Covering Games: Approximation through Non-Cooperation
Martin Gairing
· 12
Covering Games
Distributed (Local Search) Approximation Algorithm
Distributed Approximation Algorithm
Task
I
Turn this into (1 − 1e )-approximation algorithm.
I
I
I
Start with arbitrary strategy profile.
Let players unilaterally improve. (selfish steps)
Use Rosenthals potential function to bound running time.
Problem
I Increase in potential function can be arbitrary small.
Covering Games: Approximation through Non-Cooperation
Martin Gairing
· 12
Covering Games
Distributed (Local Search) Approximation Algorithm
Distributed Approximation Algorithm
Task
I
Turn this into (1 − 1e )-approximation algorithm.
I
I
I
Start with arbitrary strategy profile.
Let players unilaterally improve. (selfish steps)
Use Rosenthals potential function to bound running time.
Problem
I Increase in potential function can be arbitrary small.
Solution
I choose constant k 0 ∈ N
I
f (i) = 0 for i > k 0
I
This yields (1 −
1
e
− ε)-approximation algorithm (ε = ε(k 0 ) = o(1))
Covering Games: Approximation through Non-Cooperation
Martin Gairing
· 12
Covering Games
Distributed (Local Search) Approximation Algorithm
A local search approximation algorithm
Theorem
For every ε > 0, there exists a (local-search) approximation algorithm
I
I
with approximation ratio 1 −
that uses at most
O( 1ε
1
e
− ε,
· log log( 1ε )) · W selfish steps.
Best Possible [Feige, JouACM’98]
Covering Games: Approximation through Non-Cooperation
Martin Gairing
· 13
Covering Games
Distributed (Local Search) Approximation Algorithm
A local search approximation algorithm
Theorem
For every ε > 0, there exists a (local-search) approximation algorithm
I
with approximation ratio 1 −
O( 1ε
1
e
− ε,
· log log( 1ε )) · W selfish steps.
I
that uses at most
I
What happens if W is arbitrary?
Theorem
Then, for every (non-constant) utility sharing function, computing a
pure Nash equilibrium is PLS-complete.
Covering Games: Approximation through Non-Cooperation
Martin Gairing
· 13
Covering Games
Distributed (Local Search) Approximation Algorithm
A local search approximation algorithm
Theorem
For every ε > 0, there exists a (local-search) approximation algorithm
I
with approximation ratio 1 −
O( 1ε
1
e
− ε,
· log log( 1ε )) · W selfish steps.
I
that uses at most
I
What happens if W is arbitrary?
Theorem
Then, for every (non-constant) utility sharing function, computing a
pure Nash equilibrium is PLS-complete.
Theorem
There exists a (centralized) polynomial-time (1 − 1e )-approximation
algorithm for the general covering problem.
Covering Games: Approximation through Non-Cooperation
Martin Gairing
· 13
Conclusion
Overview
Covering Games
We showed:
I
For every utility sharing function f , PoAf ≤ 1 − e1 .
I
There exists f with PoAf ≥ 1 − 1e .
I
Local search approximation algorithm if W is bounded by polynom
in n, |E|.
I
Limits of our approach
I
Centralized Approximation Algorithm
Covering Games: Approximation through Non-Cooperation
5
3
4
6
11
9
2
1
9
6
2
13
24
8
9
18
6
3
1
1
Martin Gairing
· 14
Conclusion
Overview
Covering Games
We showed:
I
For every utility sharing function f , PoAf ≤ 1 − e1 .
I
There exists f with PoAf ≥ 1 − 1e .
I
Local search approximation algorithm if W is bounded by polynom
in n, |E|.
I
Limits of our approach
I
Centralized Approximation Algorithm
Open Problems
I
I
weighted case: restrict to ε-NE
More general models
I
I
I
we is not constant
element must be covered multiple times
...
Covering Games: Approximation through Non-Cooperation
5
3
4
6
11
9
2
1
9
6
2
13
24
8
9
18
6
3
1
1
Martin Gairing
· 14