[PDF] - Self ish Rout ing Tim Roughgarden Cornell Universit y Includes PDF Document

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Self ish Rout ing

Tim Roughgarden Cornell Universit y

Includes joint work with Éva Tardos

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Traf f ic in Congest ed Net works

The Model:

A dir ect ed gr aph G = (V,E)
k source-dest inat ion pair s

(s1 ,t 1 ), … , (sk ,t k )

A rat e r i of t r af f ic f r om si t o t i
For each edge e, a lat ency

f unct ion l e(•)

s1 t 1 l (x)=x Example: (k,r=1)

Flow = ½ Flow = ½

l (x)=1

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Flows and t heir Cost

Traf f ic and Flows:

f P = amount of t raf f ic rout ed on si-t i

pat h P

f low vect or f

rout ing of t raf f ic

The Cost of a Flow:

l P(f ) = sum of lat encies of edges on

P (w.r.t . t he f low f )

C(f ) = cost or t ot al lat ency of f low f :

ΣP f P • l P(f ) s t

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Self ish Rout ing

f low = rout es of many

noncooper at ive agent s

Examples:

– cars in a highway syst em [Wardrop 52] – packet s in a net work

cost (t ot al lat ency) of a f low

as a measur e of social welf are

agent s ar e self ish

– do not care about social welf are – want t o minimize personal lat ency

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Flows at Nash Equilibr ium

x

s t

1

Flow = .5 Flow = .5

s t

1

Flow = 0 Flow = 1

x

Def : A f low is at Nash equilibrium (is

a Nash f low) if no agent can improve it s lat ency by changing it s pat h

t his f low is envious!

Assumpt ion: edge lat ency f unct ions

are cont inuous, nondecreasing

Lemma: f is a Nash f low all f low

n minimum-lat ency pat hs (w.r.t . f )

Fact : have exist ence, uniqueness

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The I nef f iciency of Nash Flows

Fact : Nash f lows do not opt imize

t ot al lat ency [Pigou 1920] ⇒ lack of coordinat ion leads t o

inef f iciency s t x 1

1

½ ½ Cost of Nash f low = 1•1 + 0•1 = 1 Cost of opt imal (min-cost ) f low = ½

½

+½

1 = ¾

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How Bad is Self ish Rout ing?

How inef f icient are Nash f lows:

wit h mor e r ealist ic lat ency f ns?
in more realist ic net wor ks?

Goal: pr ove t hat Nash f lows ar e near-opt imal

want a laissez-f aire approach t o

managing net works

– also [Kout soupias/ P apadimit riou 99]

s t x 1

1

½ ½

Pigou’s example is simple…

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The Bad News

Bad Example: (r = 1, d large) Nash f low has cost 1, min cost ≈ 0

⇒ Nash f low can cost ar bit r ar ily

more t han t he opt imal (min- cost ) f low

– even if lat ency f unct ions are polynomials s t xd 1

1 1-?

?

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A Bicrit eria Bound

Approach # 1: set t le f or weaker t ype of guar ant ee

Theorem: [Roughgarden/ Tardos 00]

net wor k w/ ct s, nondecr easing lat ency f unct ions ⇒ = Corollary: M/ M/ 1 delay f ns (l (x)=1/ (u-x), u = capacit y) ⇒ = cost of Nash at rat e r cost of opt at rat e 2r Nash cost w/ capacit ies 2u

pt cost w/

capacit ies u

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Linear Lat ency Funct ions

Approach # 2: rest rict class of allowable lat ency f unct ions

Def : a linear lat ency f unct ion is

f t he f orm l e(x)=aex+be

Theorem: [Roughgarden/ Tardos 00]

net wor k w/ linear lat ency f ns ⇒

= 4/ 3 × cost of Nash f low cost of

pt f low

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Sources of I nef f iciency

Cor ollar y of main Theor em:

For linear lat ency f ns, worst

Nash/ OPT rat io is realized in a t wo-link net wor k!

one source of inef f iciency:

– conf ront ed w/ t wo rout es, self ish users overcongest one of t hem

Corollary ⇒ t hat ' s all, f olks!

– net work t opology plays no role s t x 1

1

½ ½

Cost of Nash = 1
Cost of OP

T = ¾

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No Dependence on Net work Topology

Theor em: [Roughgar den 02] f or (almost ) any class of lat ency f ns including t he const ant f ns, worst Nash/ OPT rat io occurs in a t wo-link net wor k. Corollary: wor st -case f or bounded-degr ee polynomials is:

s t xd 1

1 1-?

?

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Coping wit h Self ishness

Mot ivat ion:

Nash f lows inef f icient
cent ralized rout ing of t en

inf easible Goal: design/ manage net wor ks s.t . self ish rout ing “not t oo bad” ⇒ adds new algor it hmic dimension Two Approaches:

net work design (next )
St ackelberg rout ing (see t hesis)

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Braess’s Paradox

Bet t er net wor k, wor se Nash f low: Cost of Nash f low = 1.5 Cost of Nash f low = 2

s t x 1 x 1 s t x 1 ½ x 1 ½ ½ ½

r at e = 1

All t r af f ic exper iences addit ional lat ency! [Br aess 68]

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Designing Net works f or Self ish Users

The Problem:

given net wor k G = (V,E,l )

– assume single-commodit y

f ind subnet wor k minimizing

lat ency exper ienced by all self ish users in a Nash f low

s t

x 1 x 1

s t

x 1 x 1

⇒ want t o avoid Braess’s Paradox

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Generalizing Braess’s Par adox

Quest ion: is Braess’s Paradox mor e sever e in bigger net wor ks? Fact : wit h linear lat ency f ns, worst case is Reason: wit h linear lat ency f ns,

average lat ency

f Nash f low

average lat ency

f any ot her f low

= 4/ 3 ×

s t x 1 x 1 s t x 1 x 1

vs.

cost = 3/ 2 cost = 2

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Braess’s Paradox wit h General Lat ency Fns

s t s t

Nash in whole graph common lat ency = 4 Nash in opt subgraph common lat ency = 1

A Bigger Br aess Par adox: ⇒ removing edges can improve Nash by a n/ 2 f act or (n=| V|) Thm: [R 01] t his is wor st possible.

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The Tr ivial Algor it hm

Def : The t r ivial algor it hm is t o

build t he ent ir e net wor k. We know: t he t rivial algorit hm is

a 4/ 3-appr ox alg wit h linear

lat ency f ns

an n/ 2-appr ox alg wit h gener al

lat ency f ns Quest ion: what about mor e sophist icat ed algor it hms?

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Designing Net works f or Self ish Users is Hard

Thm: [R 01] For ? > 0, no (n/ 2 - ?)- appr oximat ion algor it hm exist s (unless P=NP). Thm: [R 01] For linear lat ency f unct ions, no (4/ 3 - ?)-appr ox algor it hm exist s (unless P=NP). Remar k: similar result s hold f or

t her classes of lat ency f ns.

Corollary: Br aess’s Par adox eludes ef f icient algorit hms.