Greedy Selfish Network Creation - Semantic Scholar

Introduction

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Max-Version

Greedy Selfish Network Creation Pascal Lenzner Humboldt-Universit¨ at zu Berlin

8th Workshop on Internet & Network Economics 10 December 2012

Conclusion

Introduction

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Max-Version

Model of Selfish Network Creation (1) • n selfish agents want to create a connected network

• agents want to minimize cost for network usage while

maximizing connection quality

Network Creation Game

[Fabrikant et al., PODC’03]

• actions: buy, remove, or swap own incident edges • each edge costs α > 0

• (G , α) is network induced by all agents’ strategies • cost of an agent in network (G , α):

cost(u) = edgecost(u) + distancecost(u) • edgecost(u) = α · (#edges bought by agent u)

Conclusion

Introduction

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Conclusion

Model of Selfish Network Creation (2) Two versions of each model, depending on distance-cost:

Sum-Version:


(P distancecost(u) =

Max-Version:

v ∈V (G ) dG (u, v ),

∞,

if (G , α) is connected otherwise

[Demaine et al., PODC’07]

( maxv ∈V (G ) dG (u, v ), if (G , α) is connected distancecost(u) = ∞, otherwise • pure strategy Su of agent u:

Su = set of endpoints of edges bought by agent u

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Conclusion

Previous Work Network Creation Game • Solution Concept:

pure Nash Equilibrium (NE)

• Price of Stability: O(1) [Fabrikant et al., PODC’03]

• Price of Anarchy:

α ∈ o(n) ∨ α > 273n: O(1) [Demaine+ , PODC’07] & [Mihal´ ak+ , SAGT’10]

α ∈ Θ(n):

√ 2O( log n)

[Demaine et al., PODC’07]

• computing a best response:

NP-hard


Swap-Versions • Solution Concept: • Swap Equilibrium [Alon et al., SPAA’10]

• Asymmetric Swap

Equilibrium [Mihal´ ak & Schlegel, MFCS’12]

• Price of Stability: O(1) √

• PoA: 2O( log n)

• computing a best response:

by brute-force in O(n2 ) [Alon et al., SPAA’10]

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Conclusion

Modelling Decentralized Network Creation • original motivation:

Modelling creation/evolution of Internet-like networks (agents = ISPs, strategy = links to other ISPs) • Swap-Versions: • BR computation is easy but networks cannot grow • Original Model: • highly flexible networks but BR computation is hard

• suitable model for analyzing Internet-like networks • NE is unrealistic solution concept for poly-time agents

⇒ Consider NCGs with more realistic solution concept

• agents prefer smooth strategy-adaptation over radical changes • check for improving strategy in poly-time

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Conclusion

Greedy Agents • greedy agents check three simple infrastructure improvements: • greedy augmentation: create one new own link • greedy deletion: remove one own link • greedy swap: swap one own link

• naive algorithm for checking improvement in O(n2 (n + m))

⇒ new solution concept:

Greedy Equilibrium (GE) A network (G , α) is in Greedy Equilibrium if no agent can strictly decrease her cost by buying, deleting or swapping one own edge. • GE are solutions found by distributed local search

Introduction

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Quality of Greedy Equilibria - Sum-Version Remember: cost(u) = edgecost(u) +

P

w ∈V (G ) dG (u, w )

Question: How much stability is lost by agents being greedy?

Results (Sum-Version): • Being greedy is optimal on tree networks

• Every GE-network is in 3-approximate Nash Equilibrium • Lower bound of 32 on approximation ratio

Conclusion

Introduction

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Tree Networks in GE (1) Some observations: • if agent u cannot swap own edge {u, w } to improve, then w

must be a 1-median vertex of the respective subtree

Definition: 1-median vertex of connected graph G x is 1-median of G , if x ∈ arg minu∈V (G )

P

w ∈V (G ) d(u, w ).

Conclusion

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y

x u

T∗

P

w ∈V (G ) d(u, w ).

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Definition: 1-median vertex of connected graph G x is 1-median of G , if x ∈ arg minu∈V (G ) dist(y) = 15 y

x

dist(x) = 14 u

T∗

P

w ∈V (G ) d(u, w ).

Conclusion

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Definition: 1-median vertex of connected graph G x is 1-median of G , if x ∈ arg minu∈V (G ) Ty y

x u

T∗

P

w ∈V (G ) d(u, w ).

Conclusion

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x u

T∗

P

w ∈V (G ) d(u, w ).

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x u

T∗

P

w ∈V (G ) d(u, w ).

Conclusion

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y

x u

T

P

w ∈V (G ) d(u, w ).

Conclusion

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y

x u

T∗

P

w ∈V (G ) d(u, w ).

Conclusion

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y

x u

T∗

P

w ∈V (G ) d(u, w ).

Conclusion

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Conclusion

Tree Networks in GE (2) • greedy strategy of agent u has exactly one edge per connected

component of T ∗ which does not contain u

Key Observation for Tree Networks in GE: If agent can non-greedily decrease her cost, then the new strategy must buy more edges than the current strategy. • Problem: Buying more than one edge may be optimal!

Introduction

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α=6

3

1 3

3 2

3

x

1

4 u

x1 1

2

2 3

2

4

2

2 3

cost(u) = α + 41

4 4

2

x2

2 2

x 2

2 u

2 1

cost(u) = 3α + 27

x3 2 2

Introduction

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Conclusion

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α=6

3

1 3

3 2

3

x

1

4 u

x1 1

2

2 3

2

4

2

2 3

cost(u) = α + 41

2 2

x 2

1 cost(u) = 3α + 27

2 u

2

4 4

2

x2

x3 2 2

Introduction

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Conclusion

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α=6

3

1 3

3 2

3

x

1

4 u

x1 1

2

2 3

2

4

2

2 3

cost(u) = α + 41

4 4

2

x2

3 2

x 2

3 u

2 1

cost(u) = 3α + 27

x3 2 2

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Tree Networks in GE (3) Theorem 1 If (T , α) is a tree network in GE, then (T , α) is in NE.

Conclusion

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Conclusion

Tree Networks in GE (3) Theorem 1 If (T , α) is a tree network in GE, then (T , α) is in NE. Proof: We show: If agent u can improve by any strategy change, then there is an agent z who can improve by greedy move.

Introduction

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Conclusion

Tree Networks in GE (3) Theorem 1 If (T , α) is a tree network in GE, then (T , α) is in NE. Proof: We show: If agent u can improve by any strategy change, then there is an agent z who can improve by greedy move. • if improvement by greedy strategy change, then trivially true

⇒ may assume, that u cannot improve greedily

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Conclusion



• by observation: agent u must buy strictly more edges

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• by pigeonhole principle: u must buy at least two edges to

some connected component of T ∗ which does not contain u

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• by pigeonhole principle: u must buy at least two edges to

some connected component of T ∗ which does not contain u

• we focus on this component and find agent z within

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Conclusion

Tree Networks in GE (3) Theorem 1 If (T , α) is a tree network in GE, then (T , α) is in NE. Proof: We show: If agent u can improve by any strategy change, then there is an agent z who can improve by greedy move.

x

u

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Conclusion

Tree Networks in GE (3) Theorem 1 If (T , α) is a tree network in GE, then (T , α) is in NE. Proof: We show: If agent u can improve by any strategy change, then there is an agent z who can improve by greedy move. • consider agent u’s smallest best new

strategy S ∗ within component

x

u

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x

u

• x ∈ / S ∗ and |S ∗ | ≥ 2

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x

u

• x ∈ / S ∗ and |S ∗ | ≥ 2

Lemma If u cannot improve by buying 1 edge, then u cannot improve by buying k edges.

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Conclusion

Tree Networks in GE (3) Theorem 1 If (T , α) is a tree network in GE, then (T , α) is in NE. Proof: We show: If agent u can improve by any strategy change, then there is an agent z who can improve by greedy move. Ta

• consider agent u’s smallest best new

Tb


b a

u

x c Tc

• x ∈ / S ∗ and |S ∗ | ≥ 2

• not all new edges can point into same

subtree of 1-median vertex x

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b a

u

x c

• x ∈ / S ∗ and |S ∗ | ≥ 2



• closest new endpoint to x is not a leaf

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b a

u

x c

• x ∈ / S ∗ and |S ∗ | ≥ 2




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b a z

u

x c

• x ∈ / S ∗ and |S ∗ | ≥ 2




⇒ z is neighbor of closest endpoint to x

Introduction

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Conclusion

Tree Networks in GE (3) Theorem 1 If (T , α) is a tree network in GE, then (T , α) is in NE. Proof: We show: If agent u can improve by any strategy change, then there is an agent z who can improve by greedy move. • removing {u, b} strictly increases

agent u’s cost (since S ∗ is minimal)

b a z

u

x c

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b a z

u

x c

⇒ {u, b} yields distance decrease of more than α to agent u

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b a z

u

x c

⇒ {u, b} yields distance decrease of more than α to agent u • in (T , α) agent z in similar situation

as agent u with S ∗ but without edge {u, b}

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b a z

u

x c

⇒ {u, b} yields distance decrease of more than α to agent u • in (T , α) agent z in similar situation

as agent u with S ∗ but without edge {u, b}

⇒ agent z can buy edge {z, b} in (T , α) and strictly decrease cost

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Non-Tree Networks in GE (1) Tree Networks

NE

GE

Conclusion

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N E = GE

Conclusion

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Non-Tree Networks

NE N E = GE (H2 , 3)

Conclusion

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Non-Tree Networks

NE N E = GE (H2 , 3)

(H3 , 3)

Conclusion

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Non-Tree Networks

NE N E = GE (H2 , 3)

(H3 , 3) cost = 2α + 6

Conclusion

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Non-Tree Networks

NE N E = GE (H2 , 3)

(H3 , 3) cost = 2α + 6 cost = α + 8

Conclusion

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Conclusion


Non-Tree Networks

NE

GE

N E = GE (H2 , 3)

(H3 , 3)

Theorem 2 There are non-tree networks in GE, which are not in β-approximate NE for β < 32 .

Introduction

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Non-Tree Networks

NE

GE 3-approx. NE

N E = GE (H2 , 3)

(H3 , 3)

Theorem 2 There are non-tree networks in GE, which are not in β-approximate NE for β < 32 .

Theorem 3 Every network in GE is in 3-approximate NE.

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Non-Tree Networks in GE (2) Theorem 3 Every network in GE is in 3-approximate NE.

Conclusion

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Non-Tree Networks in GE (2) Theorem 3 Every network in GE is in 3-approximate NE. Proof: By “locality-gap preserving” reduction to Uncapacitated Metric Facility Location (UMFL).

Conclusion

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Non-Tree Networks in GE (2) Theorem 3 Every network in GE is in 3-approximate NE. Proof: By “locality-gap preserving” reduction to Uncapacitated Metric Facility Location (UMFL). d e

f

c a

b

u

Network (G, α)

Conclusion

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f

c a

b

u

Network (G0 , α)

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a

b

c

d

e

f

Clients

a

b

c

d

e

f

Facilities

f

c a

b

u

Network (G0 , α)

UMFL instance I(G0 )

Introduction

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Non-Tree Networks in GE (2) Theorem 3 Every network in GE is in 3-approximate NE. Proof: By “locality-gap preserving” reduction to Uncapacitated Metric Facility Location (UMFL). a

d e

f

3 2

1

c a

c

b 2

2

d 1

3 2

2

e 1

2 3 3

1

1

f

Clients

1 2

b

u

Network (G0 , α)

a

2

2

b

c

d

e

UMFL instance I(G0 )

2

f

Facilities

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d e

f

3 2

1

c a

c

b 2

2

d 1

3 2

2

e 1

2 3 3

1

1

f

Clients

1 2

b

u

a α

Network (G0 , α)

2

2

b 0

c

d

e

α α α UMFL instance I(G0 )

2

f

Facilities

α

Opening Cost

Introduction

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d 2

c

b

e c

2

f

1

2 1

2

2

d 1

3 2

2

e 1

2 3 3

1

1

f

Clients

1 2

1 a

3

2

b

1

u

a α

Network (G0 , α)

2

2

b 0

c

d

e


2

f

Facilities

α

Opening Cost

• Bijection between agent u’s strategies and UMFL solutions

• same cost of u’s strategy and corresponding UMFL solution

Introduction

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Conclusion

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d 2

c

b

e c

2

f

1

2 1

2

2

d 1

3 2

2

e 1

2 3 3

1

1

f

Clients

1 2

1 a

3

2

b

1

u

a α

Network (G0 , α)

2

2

b 0

c

d

e


2

f

Facilities

α

Opening Cost

• Bijection between agent u’s strategies and UMFL solutions

• same cost of u’s strategy and corresponding UMFL solution • Arya et al. [STOC’01]: UMFL has locality-gap of 3.

Introduction

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Max-Version

Quality of Greedy Equilibria - Max-Version Remember: cost(u) = edgecost(u) + maxw ∈V (G ) dG (u, w )

Question: How much stability is lost by agents being greedy?

Results (Max-Version): • Being greedy is sub-optimal on tree networks, but • “bad” trees can be characterized (and checked in poly-time) • if diameter at most 2, then (T , α) in 2-approximate NE • if diameter at least 3, then (T , α) in 65 -approximate NE • both bounds are tight • Worse situation for non-tree networks: • Ω(n) lower bound on approximation ratio ⇒ locality-gap of Ω(n) for Min-Max Metric Facility Location

Conclusion

Introduction

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Max-Version

Non-Tree Networks in GE Theorem 4 For 1 ≤ α ≤ 2 there is a n vertex network (G , α) in GE, which is not in β-approximate NE for any β < n−1 5 .

Conclusion

Introduction

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Conclusion

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Non-Tree Networks in GE Theorem 4 For 1 ≤ α ≤ 2 there is a n vertex network (G , α) in GE, which is not in β-approximate NE for any β < n−1 5 . Proof: a1

u

u

a2

x1

x2

...

x3

l2

l1 b1

b2 v

a1

a2

y1

y2

...

y3

x1

x2

...

x3

y1

y2

...

y3

l2

l1 b1

b2 v

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u

u

a2

x1

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...

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l2

l1 b1

b2 v

a1

a2

y1

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...

y3

x1

x2

...

x3

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...

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l1 b1

b2 v

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u

u

a2

x1

x2

...

x3

l2

l1 b1

b2 v

a1

a2

y1

y2

...

y3

x1

x2

...

x3

y1

y2

...

y3

l2

l1 b1

b2 v

cost(u) 3α+3 7 2 9 • for α = 2 this yields: cost ∗ (u) = α+3 = 5 + 5 = 5

Introduction

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Conclusion

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u

u

a2

x1

x2

...

x3

l2

l1 b1

b2 v

a1

a2

y1

y2

...

y3

x1

x2

...

x3

y1

y2

...

y3

l2

l1 b1

b2 v

α(2+2)+3 cost(u) • for α = 2 this yields: cost = 75 + 45 = 11 ∗ (u) = α+3 5

Introduction

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Conclusion

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u

u

a2

x1

x2

...

xk

l2

l1 b1

b2 v

a1

a2

y1

y2

...

yk

x1

x2

...

xk

y1

y2

...

yk

l2

l1 b1

b2 v

α(2+k)+3 cost(u) • for α = 2 this yields: cost = 75 + 2k ∗ (u) = α+3 5

Introduction

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Conclusion

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u

u

a2

x1

x2

...

xk

l2

l1 b1

b2

a1

a2

y1

y2

...

yk

x1

x2

...

xk

y1

y2

...

yk

l2

l1 b1

v

b2 v

α(2+k)+3 cost(u) • for α = 2 this yields: cost = 75 + 2k ∗ (u) = α+3 5 cost(u) n−1 • since k = n−8 2 , we have cost ∗ (u) = 5

Introduction

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Max-Version

Summary of Results

Results for tree networks in GE • Greediness suffices to create stable trees in Sum-Version

• Greediness almost optimal in Max-Version • Nash stability can be checked in poly-time

Results for non-tree networks in GE • Every GE is a 3-approximate NE in Sum-Version

• GE in Max-Version may only be in Ω(n)-approximate NE

Conclusion

Introduction

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e

u

s

Q

t i o n s

?

Conclusion