# Braess’s Paradox, Fibonacci Numbers, and Exponential Inapproximability - PowerPoint PPT Presentation

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Braess’s Paradox, Fibonacci Numbers, and Exponential Inapproximability. Henry Lin * Tim Roughgarden ** Éva Tardos † Asher Walkover †† * UC Berkeley ** Stanford University † Cornell University †† Google. Overview. Selfish routing model and Braess’s Paradox

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Braess’s Paradox, Fibonacci Numbers, and Exponential Inapproximability

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## Braess’s Paradox, Fibonacci Numbers, and Exponential Inapproximability

Henry Lin*

Tim Roughgarden**

Éva Tardos†

Asher Walkover††

*UC Berkeley**Stanford University

### Overview

• Selfish routing model and Braess’s Paradox

• New lower and upper bounds on Braess’s Paradox in multicommodity networks

• Connections to the price of anarchy with respect to the maximum latency objective

• Open questions

### Routing in congested networks

• a directed graph: G = (V,E)

• for each edge e, a latency function: ℓe(•)

• nonnegative, nondecreasing, and continuous

• one or more commodities: (s1, t1, r1) … (sk, tk, rk)

• for i=1 to k, a rate ri of traffic to route from si to ti

Single Commodity Example (k=1):

r1=1

v

ℓ(x)=1

ℓ(x)=x

Flow = ½

s1

t1

ℓ(x)=x

Flow = ½

ℓ(x)=1

u

### Selfish Routing and Nash Flows

How do we model selfish behavior in networks?

Def: A flow is at Nash equilibrium (or is a Nash flow) if all flow is routed on min-latency paths

[at current edge congestion]

• Note: at Nash Eq., all flow must have same s to t latency

• Always exist & are unique [Wardrop, Beckmann et al 50s]

An example Nash flow:

v

k=1, r1=1

ℓ(x)=1

ℓ(x)=x

Flow = ½

s1

t1

ℓ(x)=x

Flow = ½

ℓ(x)=1

u

• Common latency is 1.5

• Adding edge increased latency to 2!

• Replacing x with xd yields more severe example where latency increases from 1 to 2

v

½

1

½

1

x

0

s

t

½

1

½

x

1

u

### Previous results on Braess’s Paradox

In single-commodity networks:

• Thm: [R 01]Adding 1 edge to a graph can increase common latency by at least a factor of 2

• Thm: [LRT 04]Adding 1 edge to a graph can increase common latency by at most a factor of 2

### New results for BPin multicommodity networks

In a network with k ≥ 2 commodities, n nodes, m edges:

• Thm: Adding 1 edge to a graph can increase common latency by at least a factor of 2Ω(n) or 2Ω(m), even if k = 2

• Thm: Adding 1 edge to a graph can increase common latency at most a factor of 2O(m·logn)or 2O(kn),whichever is smaller

### Braess’s Paradox in MC networks

t2

r1 = r2 = 1

• All unlabelled edges have 0 latency (at current flow)

• Only edge leaving s1 has latency 1

• Latency between s1 and t1 is 1

• Latency between s2 and t2 is 0

1

s1

t1

s2

### Braess’s Paradox in MC networks

t2

r1 = r2 = 1

• All unlabelled edges have 0 latency (at current flow)

1

s1

t1

1

-½ flow

+½ flow

s2

### Braess’s Paradox in MC networks

t2

r1 = r2 = 1

• All unlabelled edges have 0 latency (at current flow)

1

s1

t1

1

1

-¼ flow

+¼ flow

s2

### Braess’s Paradox in MC networks

t2

r1 = r2 = 1

• All unlabelled edges have 0 latency (at current flow)

1

1

s1

t1

1

1

-⅛ flow

+⅛ flow

s2

### Braess’s Paradox in MC networks

t2

r1 = r2 = 1

• All unlabelled edges have 0 latency (at current flow)

1

1

s1

t1

2

1

1

-1/16 flow

+1/16 flow

s2

### Braess’s Paradox in MC networks

t2

r1 = r2 = 1

• All unlabelled edges have 0 latency (at current flow)

3

-1/32 flow

+1/32 flow

1

1

s1

t1

2

1

1

s2

### Braess’s Paradox in MC networks

t2

-1/64 flow

+1/64 flow

3

1

1

s1

t1

2

5

1

1

s2

• All unlabelled edges have 0 latency (at current flow)

### Braess’s Paradox in MC networks

t2

8

-1/128 flow

+1/128 flow

3

1

1

s1

t1

2

5

1

1

s2

• All unlabelled edges have 0 latency (at current flow)

### Braess’s Paradox in MC networks

t2

• Latency between s1 and t1 increased from 1 to 9

• Latency between s2 and t2 increased from 0 to 13

8

3

1

1

s1

t1

2

5

1

1

s2

• All unlabelled edges have 0 latency (at current flow)

t2

8

3

1

1

s1

t1

2

5

1

1

s2

### Braess’s Paradox in MC networks

• In a general network with O(p) nodes:

• Latency between s1 and t1 can increase from 1 to Fp-1+1

• Latency between s2 and t2 can increased from 0 to Fp

• (where Fp is the pth fibonacci number)

• In fact, adding 1 edge is enough to cause this bad example

### Proving Upper Bounds

To prove 2O(m·logn)bound, let:

f be the flow before edges were added

g be the flow after edges were added

Main Lemma: For any edge e:

ℓe(ge) ≤ 2O(m·logn)·maxe’єE(ℓe’(fe’))

### Proving Main Lemma

Main Lemma: For any edge e:

ℓe(ge) ≤ 2O(m·logn)·maxe’єE(ℓe’(fe’))

Proof (sketch):Let f, g, and ℓe(fe) be fixed.

Resulting latencies ℓe(ge) must be:

• nonnegative

• nondecreasing

• at Nash equilibrium

Requirements can be formulated as a set of linear constraints on ℓe(ge)

### Proving Main Lemma

Main Lemma: For any edge e:

ℓe(ge) ≤ 2O(m·logn)·maxe’єE(ℓe’(fe’))

Proof (sketch):Let f, g, and ℓe(fe) be fixed.

In fact, finding maximum ℓe(ge) can be formulated as a linear program

• can show maximum occurs at extreme point

• can bound extreme point solution with Cramer’s rule and a bound on the determinant

### Price of Anarchy with respect to Maximum Latency Objective

• The maximum si-ti latency at Nash Eq. is 2Ω(n)

• An optimal flow avoiding the extra edges can have maximum si-ti latency equal to 1

New Thm:The price of anarchy wrt to the maximum latency is at least 2Ω(n).

Disproves conjecture that PoA for multicommodity networks is no worse than for single-commodity networks

### Price of Anarchy with respect to Maximum Latency Objective

• Linear programming technique not specific to Braess’s Paradox

• Provides same bound for price of anarchy wrt maximum latency

New Thm:The price of anarchy wrt to the maximum latency is at most 2O(m·logn) or 2O(kn), whichever is smaller

### Open Questions

• Can the upper bounds be improved to 2O(n)or 2O(m)?

• Can the lower bounds be improved to 2Ω(m·logn) or 2Ω(kn)?

• What are upper and lower bounds on Braess’s Paradox and price of anarchy for atomic splittable instances?