On a Non-Cooperative Model for Wavelength Assignment in Multifiber Optical Networks E. Bampas, A. Pagourtzis, G. Pierrakos, K. Potika {ebamp,pagour,gpier,epotik}@cs.ntua.gr National Technical University of Athens 1/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Transparent all-optical networks Much more bandwidth than legacy copper wire No opto-electronic conversion faster cheaper Wavelength Division Multiplexing (WDM) several “channels” per fiber 2/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Transparent all-optical networks Much more bandwidth than legacy copper wire No opto-electronic conversion faster cheaper Wavelength Division Multiplexing (WDM) several “channels” per fiber Multi-fiber setting fault-tolerance even more bandwidth 2/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Non-cooperative model Large-scale networks: shortage of centralized control provide incentives for users to work for the social good Social good: minimize fiber multiplicity Charge users according to the maximum fiber multiplicity incurred by their choice of frequency and/or routing 3/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Non-cooperative model Large-scale networks: shortage of centralized control provide incentives for users to work for the social good Social good: minimize fiber multiplicity Charge users according to the maximum fiber multiplicity incurred by their choice of frequency and/or routing What will be the impact on social welfare if we allow users to act freely and selfishly? 3/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Problem formulation Def. P ATH M ULTICOLORING problems: input: graph G ( V, E ) , path set P , # colors w solution: a coloring c : P → W , W = { α 1 , . . . , α w } goals: minimize the sum of maximum color multiplicities e ∈ E max α ∈ W µ ( e, α ) [NPZ01], or � minimize the maximum color multiplicity µ max � max e ∈ E max α ∈ W µ ( e, α ) [AZ04] 4/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Problem formulation Def. P ATH M ULTICOLORING problems: input: graph G ( V, E ) , path set P , # colors w solution: a coloring c : P → W , W = { α 1 , . . . , α w } goal: minimize the maximum color multiplicity µ max � max e ∈ E max α ∈ W µ ( e, α ) p 1 e 2 L ( e 2 ) = 2 µ e 1 = 2 e 1 L = 3 µ max = 2 µ ( e 2 , green) = 1 µ ( p 1 , blue) = 2 4/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Problem formulation Def. P ATH M ULTICOLORING problems: input: graph G ( V, E ) , path set P , # colors w solution: a coloring c : P → W , W = { α 1 , . . . , α w } goal: minimize the maximum color multiplicity µ max � max e ∈ E max α ∈ W µ ( e, α ) p 1 e 2 µ OPT = 1 e 1 � L � µ OPT ≥ w 4/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Game-theoretic formulation Def. Given a graph G , path set P and w , define the game � G, P, w � : players: p 1 , . . . , p | P | ∈ P strategies: each p i picks a color c i ∈ W strategy profile: a vector � c = ( c 1 , . . . , c | P | ) disutility functions: for p i ∈ P , f i ( � c ) = µ ( p i , c i ) social cost: c ) � µ max = max sc( � e ∈ E max α ∈ W µ ( e, α ) 5/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Game-theoretic formulation Def. Given a graph G , path set P and w , define the game � G, P, w � : players: p 1 , . . . , p | P | ∈ P strategies: each p i picks a color c i ∈ W strategy profile: a vector � c = ( c 1 , . . . , c | P | ) disutility functions: for p i ∈ P , f i ( � c ) = µ ( p i , c i ) social cost: c ) � µ max = max sc( � e ∈ E max α ∈ W µ ( e, α ) Def. S-PMC: the class of all � G, P, w � games 5/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Nash Equilibria Def. A strategy profile is a Nash Equilibrium (NE) if no player can reduce her disutility by changing strategy unilaterally: ∀ p i ∈ P, ∀ c ′ c ; c ′ i ∈ W : f i ( � c ; c i ) ≤ f i ( � i ) Def. ε -approximate Nash Equilibrium: no player can reduce her disutility by more than a factor of 1 − ε Def. We denote the social cost of the worst-case NE by ˆ µ : µ = max ˆ c is NE sc( � c ) � 6/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Efficiency of Nash Equilibria Def. The price of anarchy (PoA) of an S-PMC game: ˆ µ PoA = µ OPT Def. The price of stability (PoS) of an S-PMC game: PoS = min � c is NE sc( � c ) µ OPT 7/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Efficiency of Nash Equilibria Def. The price of anarchy (PoA) of an S-PMC game: ˆ µ PoA = µ OPT Def. The price of stability (PoS) of an S-PMC game: PoS = min � c is NE sc( � c ) µ OPT Rate of convergence to some NE? by repeatedly changing some player’s strategy to improve her disutility (Nash dynamics) 7/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Results in this work Any Nash dynamics converges in at most 4 | P | steps Efficient computation of NE: optimal NE for S-PMC(R OOTED -T REE ) 2 -approximate NE for S-PMC(S TAR ) 1 Upper and lower bounds for the PoA: # colors minimum length of any path that contributes to the cost of some worst-case NE matching lower bounds for graphs with ∆ ≥ 3 8/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Results in this work (Cont’d) The PoA on graphs with degree 2 : if L = Ω( w 2 ) , PoA = O (1) if L = o ( w 2 ) , PoA is unbounded 9/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Related work Price of anarchy [KP99], price of stability [ADK + 04] Congestion games [MS96, Ros73] player cost: SUM of delays of selected resources large body of work Bottleneck network games player cost: MAX of delays along her path players pick among several possible routings [BM06] latency functions on edges [BO06] 10/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Convergence to NE Thm. Any Nash dynamics converges in at most 4 | P | steps consider the vector ( d L ( � c ) , d L − 1 ( � c ) , . . . , d 1 ( � c )) lexicographic-order argument (attributed to Mehlhorn in [FKK + 02]) PoS = 1 11/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Convergence to NE Thm. Any Nash dynamics converges in at most 4 | P | steps consider the vector ( d L ( � c ) , d L − 1 ( � c ) , . . . , d 1 ( � c )) lexicographic-order argument (attributed to Mehlhorn in [FKK + 02]) PoS = 1 how many such vectors? � | P | + L − 1 � ≤ 2 | P | + L − 1 < 4 | P | | P | 11/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Efficient computation of optimal NE � G, P, w � is in S-PMC(R OOTED -T REE ) if ∃ r s.t. each path in P lies entirely on some simple path from r to a leaf consider edges in BFS order: color paths with min-multiplicity color in the partial solution 12/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
A structural property of NE If � c is a NE, then for any p i ∈ P and for any α ∈ W there is an e ∈ p i s.t. µ ( e, α ) ≥ f i ( � c ) − 1 red-blocking edge for p i p i µ µ − 1 µ − 1 red-blocking paths for p i 13/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
An upper bound on the PoA Thm. If � c is a NE and sc( � µ then PoA ≤ len( p i ) c ) = f i ( � c ) = ˆ Proof. all w colors are blocked along p i � � some edge of p i must block at least colors w len( p i ) � � max load is L ≥ 1 + w (ˆ µ − 1) len( p i ) � L � µ OPT ≥ w ˆ ˆ µ µ PoA = µ OPT ≤ ≤ len( p i ) ‰ ı 2 3 w 1+ (ˆ µ − 1) len( pi ) 6 7 w 6 7 6 7 14/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
A matching lower bound w = L = len( p i ) = ˆ µ = k µ OPT = 1 PoA = k 15/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
16/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
What about graphs with degree 2? A more involved structural property: Lem. In a NE of an S-PMC(R ING ) game, ∀ edge e and ∀ α i there is an arc s.t.: ∀ α j � = α i the arc contains an edge which is an α j -blocking edge for at least half of the paths in P ( e, α i ) , and ∀ e ′ in the arc, | P ( e ′ , α i ) ∩ P ( e, α i ) | ≥ � � | P ( e,α i ) | 2 17/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
Establishing an edge with high load Repeated application of the previous Lemma yields: Lem. In every S-PMC(R ING ) game � G, P, w � with ˆ µ ≥ w there is an edge with load at least ˆ µw 4 18/22 Evangelos Bampas — ACAC 2008, University of Athens, August 25-26, 2008
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