The Price of Routing Unsplittable Flow Baruch Awerbuch Yossi Azar - PowerPoint PPT Presentation

The Price of Routing Unsplittable Flow Baruch Awerbuch Yossi Azar Amir Epstein presented by Yajun Wang (yalding@cs.ust.hk) for COMP670O Spring 2006, HKUST 1-1

Problem Formulation • Graph G = ( V, E ) and k source-destination pairs { s i , t i } • Q i denotes the set of (simple) s i − t i paths, and • Latency function f e : R + → R + • Bandwidth request ( s j , t j , w j ) w j ∈ R + • A flow is a function vector ( l j ) . l j : Q j → R + • A flow is feasible if : � Q ∈Q j l j ( Q ) = w j 2-1

Flow and Strategy • Splittable Flow l j ( Q ) ∈ [0 , w j ] • Unsplittable Flow l j ( Q ) ∈ { 0 , w j } Pure Strategies: User j selectes a single path Q ∈ Q j . Mixed Strategies: User j selectes a probability distribution { p Q,j } over Q j . 3-1

Latency for Users • Pure Strategies: Let S be the system of strategies. Let Q j be the choice of user j , and Q = ∪ j Q j . Define J ( e ) = { j | e ∈ Q} and l e = � j ∈ J ( e ) w j . Latency (per unit) of user j for select path Q (instead of Q j ): � � c Q,j = f e ( l e ) + f e ( l e + w j ) ( e ∈ Q ) ∧ ( e ∈ Q j ) ( e ∈ Q ) ∧ ( e/ ∈ Q j ) 4-1

Latency for Users • Mixed Strategies: Let S be the system of strategies with { p j } Let { X Q,j } be the set of indicator random variables: whether request j is assigned to Q . l e = � n X e,j = � Q | e ∈ Q X Q,j j =1 X e,j w j Expected latency (per unit) of user j for select path Q in S � = E [ f e ( l e ) | X Q,j = 1] c Q,j e ∈ Q n � � = E [ f e ( X e,i w i + w j )] e ∈ Q i =1 ,i � = j � = E [ f e ( l e + (1 − X e,j ) w j )] e ∈ Q 5-1

Nash Equilibrium A system S is at Nash equilibrium if and only if for every j ∈ { 1 , 2 , . . . , n } and Q, Q ′ ∈ Q j , with p Q,j > 0( Q = Q j ) c Q,j ≤ c Q ′ ,j Social cost (expected) for system S is: C ( S ) = E [ � e ∈ E f e ( l e ) l e ] Coordination Ration (Price of Anarchy) is: C ( S ) R = max S C ( S ∗ ) S takes over all Nash equilibrium(N.E), and S ∗ is the Social Optimal(S.O) solution. 6-1

Nash Equilibrium for Linear Latency Functions Theorem For linear latency functions and pure strategies, √ the worse-case coordination ratio R is at most 3+ 5 ≈ 2 . 618 2 Proof: Let Q j be the path assigned for request j in N.E. Let Q ∗ j be the path assigned for request j in S.O. � � � a e l e + b e ≤ a e l e + b e + a e ( l e + w j ) + b e e ∈ Q j ( e ∈ Q ∗ j ) ∧ ( e ∈ Q j ) ( e ∈ Q ∗ j ) ∧ ( e/ ∈ Q j ) � ≤ a e ( l e + w j ) + b e e ∈ Q ∗ j X X X X ( a e l e + b e ) w j + a e w 2 ( a e l e + b e ) w j ≤ j j e ∈ Q j j e ∈ Q ∗ j ( a e l e + b e ) w j + a e w 2 X X X X ( a e l e + b e ) w j ≤ j e ∈ E e ∈ E j ∈ J ( e ) j ∈ J ∗ ( e ) 7-1

Nash Equilibrium for Linear Latency Functions Proof (cont’): X X X X ( a e l e + b e ) w j + a e w 2 ( a e l e + b e ) w j ≤ j e ∈ E e ∈ E j ∈ J ( e ) j ∈ J ∗ ( e ) X X X w d e ) d w j = l ∗ j ≤ ( l ∗ w j = l e , e , j ∈ J ( e ) j ∈ J ∗ ( e ) j ∈ J ∗ ( e ) 2 � � ( a e l e + b e ) l ∗ e + a e l ∗ ( a e l e + b e ) l e ≤ e e ∈ E e ∈ E � � a e l e l ∗ ( a e l ∗ e + b e ) l ∗ = e + e e ∈ E e ∈ E 8-1

Nash Equilibrium for Linear Latency Functions Proof (cont’): � � � a e l e l ∗ ( a e l ∗ e + b e ) l ∗ ( a e l e + b e ) l e ≤ e + e e ∈ E e ∈ E e ∈ E sX X X 2 a e l e l ∗ a e l 2 a e l ∗ ≤ e e e Cauchy-Schwartz Inequality e ∈ E e ∈ E e ∈ E sX X ( a e l e + b e ) l e ( a e l ∗ e + b e ) l ∗ ≤ e e ∈ E e ∈ E √ � x 2 ≤ x + 1 , x 2 ≤ 3+ C ( S ) 5 x = C ( S ∗ ) 2 9-1

Nash Equilibrium for Linear Latency Functions Unweighted Demand: w j = 1 Theorem For linear latency functions, unweighted demand and pure strategies, the worse-case coordination ratio R is at most 2 . 5 Proof: X X X X ( a e l e + b e ) w j + a e w 2 ( a e l e + b e ) w j ≤ j e ∈ E e ∈ E j ∈ J ( e ) j ∈ J ∗ ( e ) X X a e l e l ∗ e + a e l ∗ e + b e l ∗ ( a e l e + b e ) l e ≤ e e ∈ E e ∈ E 10-1

Nash Equilibrium for Linear Latency Functions Proof: X X a e l e l ∗ e + a e l ∗ e + b e l ∗ ( a e l e + b e ) l e ≤ e e ∈ E e ∈ E e + 3 2 b e l e = 3 e + b e l e ) − 1 a e l 2 2( a e l 2 2 a e l 2 ( a e l e + b e ) l e ≤ e 3 e ) − 1 2 al 2 2( al e l ∗ e + al ∗ e + bl ∗ ≤ e 1 e ) + 3 e − l 2 2 a (3 l e l ∗ e + 3 l ∗ 2 b e l ∗ = e 5 2 + 3 3 ij + 3 j − i 2 ≤ 5 j 2 2 a e l ∗ 2 b e l ∗ ≤ e e 5 2( a e l ∗ e + b e ) l ∗ ≤ e 11-1

Nash Equilibrium for Linear Latency Functions Theorem For linear latency functions and mixed strategies, √ the worse-case coordination ratio R is at most 3+ 5 ≈ 2 . 618 2 Proof: � = E [ f e ( l e ) | X Q,j = 1] c Q,j e ∈ Q n � � = E [ f e ( X e,i w i + w j )] e ∈ Q i =1 ,i � = j � = E [ f e ( l e + (1 − X e,j ) w j )] e ∈ Q The change from X Q,j to X e,j does not affect the proofs. In particular, the proof of Lemma 3.4 is still correct, if we replace p Q,j − p 2 Q,j by (1 − p e,j ) p Q,j . 12-1

Nash Equilibrium for Linear Latency Functions Remarks: If we allow splittable flows, the price of anarchy is bounded by 4 3 [Roughgarden, SODA 05] Though I am doubt on this result, as the Proposition 1 there is counter intuitive to me. Unweighted demand will not achieve better ratio in mixed strategies. Because we lose the properties for integers. 13-1

Lower Bounds for Linear Latency Functions √ V φ = 1+ 5 , 1 Demands: 2 0 • User 1: ( U, V, φ ) x x • User 2: ( U, W, φ ) x U x • User 3: ( V, W, 1) • User 4: ( W, V, 1) 0 W 2 φ 2 + 2( φ + 1) 2 2 φ 2 + 2 Optimal: N.E • User 1: UWV • User 1: UV • User 2: UV W • User 2: UW • User 3: V UW • User 3: V W • User 4: WUV • User 4: WV 14-1

Nash Equilibrium for Polynomial Latency Functions Theorem For polynomial latency functions of degree d and pure and mixed strategies, the worse-case coordination ratio R is O (2 d d d +1 ) Theorem For polynomial latency functions of degree d and pure strategies, the worse-case coordination ratio R is Ω( d d/ 2 ) 15-1

Lower Bounds for Polynomial Latency Functions 0 Optimal: f ( x ) = x d No job for grup 0 Group k assigns jobs to 1 links of group k − 1 . Nash Equilibrium: l ! k ! links k l ! k − 1! jobs Group k assigns jobs to links of group k . l − 1 l − 1 l ! 1 k ! 1 d = l ! X X OPT = k ! ≈ l ! · e l k =0 k =0 l l ! l ! ( d/ 2) d · ( d/ 2) d = l ! · Ω( d d/ 2 ) k ! k d ≥ X NE = k =1 16-1

Remaining: Lower bounds for mixed strategies. Gap in the bounds of polynomial latency functions: O (2 d d d +1 ) and Ω( d d/ 2 ) . 17-1