Atomic Routing Games on Maximum Congestion Costas Busch, Malik Magdon-Ismail { buschc,magdon } @cs.rpi.edu June 20, 2006.
Outline • Motivation and Problem Set Up; • Related Work and Our Contributions; • Proof Sketches; • Wrap Up. 1
Routing Routing: coustruct “good” paths given sources and destinations. • Communication Networks – eg. Internet . • Ad-hoc Networks – eg. sensor networks. • Parallel Architectures – eg. Mesh. • . . . 2
Routing Routing: coustruct “good” paths given sources and destinations. • Communication Networks – eg. Internet . • Ad-hoc Networks – eg. sensor networks. ↓ • Parallel Architectures – eg. Mesh. • . . . 3
Motivation BILL ALICE � � BILL BILL � � � � � � � � �� �� �� �� �� �� � � � � � � � � � �� �� �� �� �� �� �� � � �� � � � �� � �� � �� �� �� �� � � � � �� �� � ALICE ALICE Which path? 4
Motivation BILL ALICE � � BILL BILL � � � � � � � � �� �� �� �� �� �� � � � � � � � � � �� �� �� �� �� �� �� � �� � � � � �� � �� � �� �� �� �� � � � � �� �� � ALICE ALICE delay = 15 sec 5
Motivation BILL ALICE � � BILL BILL � � � � � � � � �� �� �� �� �� �� � � � � � � � � � �� �� �� �� �� �� �� � �� � � � � �� � �� � �� �� �� �� � � � � �� �� � ALICE ALICE delay = 10 sec 6
Motivation BILL ALICE ������������� ������������� ����������� ����������� � � BILL BILL � � � � � � � ������������� ������������� ����������� � ����������� �� �� �� �� �� �� � � � � � � � � � ������������� ������������� ����������� ����������� �� �� �� �� �� �� �� � �� � � � � �� � �� ������������� ������������� ����������� ����������� � �� �� �� �� � � � � ������������� ������������� ����������� ����������� �� �� � ALICE ALICE ������������� ������������� ����������� ����������� delay = 10 sec 7
Routing Games • Selfish players : everyone will change paths to minimize their delay. Best Response Dynamic • Nash-Routing : no-one wishes to change her path selection, given what everyone else is doing. We study properties of this process. 8
Routing Games delay = 12 sec delay = 15 sec 1 2 delay = 5 sec 3 delay = 13 sec 4 Player cost pc i : delay of player i ’s packet. Social cost SC : maximum delay over all players. SC = 15 sec Players minimize their player cost selfishly Ideally, social cost should be minimized. 9
Quantifying Delay Congestion: C = max i C i = 3 C 4 = 3 C 2 = 3 1 2 C 3 = 2 3 C 1 = 3 4 Dilation: D = max i | p i | = 5 C i is the largest congestion on path p i . Social Cost : max i delay i = O ( C + D ) [LMR95] Player Cost : delay i = ˜ O ( C i + | p i | ) [BS95] 10
Congested Networks C ≫ D, C i ≫ | p i | Social Cost: max i delay i = O ( C ) Player Cost: delay i = ˜ O ( C i ) 11
Formal Setup Routing (Congestion) Game : ( N , G, {P i } i ∈ N ). N = { 1 , 2 , . . . , N } – players, i.e. ( source, dest ) pairs; G = ( V, E ) – network; P i – strategy sets ( edge-simple paths). Routing : p = [ p 1 , p 2 , · · · , p N ] – pure strategy profile . Congestion : C e ( p ) = # paths using edge e . Path Congestion : C i ( p ) = max e ∈ p i C e ( p ); Network Congestion : C ( p ) = max i C i ( p ); Social Cost : SC ( p ) = C ( p ) (Network Congestion). Player Cost : pc i ( p ) = C i ( p ) (Player’s Path Congestion). Nash-routing p : pc i ( p ) ≤ pc i ( p ′ ) ( p ′ differs from p only in p i ). (No one can unilaterally inprove her situation in a Nash-routing.) 12
Quality of Nash-Routings SC ( p ) Prics of Stability PoS = inf SC ∗ , p ∈ P SC ( p ) Price of Anarchy PoA = sup SC ∗ . p ∈ P PoS : minimum price for stability. (best possible selfish outcome) PoA : maximum price for stability. (worst possible selfish outcome) Ideal: PoS = PoA = 1. 13
Related Work Atomic Flow Splittable Flow Pure , , [BM06] Mixed , Max SC Sum SC Other SC Max pc – – [BM06] Sum pc , , : specific network or strategy sets (eg. parallel links or singleton sets). : existence or convergence to equilibrium (do not look at quality ( SC )). Note: sum SC is relevent when network resources, not max. player delay is important. 14
Our Contribution – PoS Routing games with max. player/social costs on general networks. Theorem 1 (i) PoS = 1; (ii) All best response dynamics converge to a Nash-routing SC ( p final ) ≤ SC ( p start ) . – There exist good Nash-routing. – Starting at any good routing, selfish players can only improve! Good oblivious starting routings: [MMVW97], [R02], [BMX05]. 15
Our Contribution – PoA Routing games with max. player/social costs on general networks. Theorem 2 PoA < 2( ℓ + log n ). ℓ upper bounds path lengths in the strategy sets. ℓ can be small (eg. Hypercubes). e + log 2 n ). Theorem 3 κ e − 1 ≤ PoA ≤ c ( κ 2 κ e ( G ) is the length of the longest cycle. PoA is bounded by topological properties of the network. 16
Proof Sketch: PoS = 1 Establish a total order ≤ c , < c among routings with: Lemma 1 There exists a minimum routing p ∗ . [Compactness of routings.] Lemma 2 SC ( p ) ≤ SC ( p ′ ) iff p ≤ c p ′ . Lemma 3 If p → p ′ in a selfish move, then p ′ < c p = ⇒ SC ( p ′ ) < SC ( p ). Corollary Minimum routings p ∗ are a Nash-routings. Best response dynamics converge to better Nash-routing. (Note: cf. potential function methods.) 17
Proof Sketch: PoA ≤ 2( ℓ + log n ) C Π 0 E 0 : Edges of congestion C . Π 0 : Players using edges in E 0 . 18
Proof Sketch: PoA ≤ 2( ℓ + log n ) C C − 1 C − 1 C − 1 Π 0 Alternative paths for players in Π 0 must all have at least one edge with congestion at least C − 1. � � E 0 : Edges of congestion C . Π 0 : Players using edges in E 0 . 19
Proof Sketch: PoA ≤ 2( ℓ + log n ) C C − 1 C − 1 C − 1 Π 1 Π 1 Π 0 E 1 : All these edges of congestion ≥ C − 1. Π 1 : Players using edges in E 1 . if | E 1 | ≤ 2 | E 0 | , stop, else continue Edge Expansion Process ( | E 0 | = 1 , | E 1 | = 4) ( E 1 is formed from all possible paths of players in Π 0 ) 20
Proof Sketch: PoA ≤ 2( ℓ + log n ) C − 2 C C − 1 C − 2 C − 1 C − 2 C − 1 Π 1 Π 1 Π 0 Alternative paths for players in Π 1 must all have at least one edge with congestion at least C − 2. � � E 1 : Edges of congestion at least C − 1. Π 1 : Players using edges in E 1 . 21
Proof Sketch: PoA ≤ 2( ℓ + log n ) C − 2 C C − 1 C − 2 C − 1 C − 2 C − 1 Π 1 Π 1 Π 0 E 2 : All these edges of congestion ≥ C − 2. if | E 2 | ≤ 2 | E 1 | , stop. ( | E 1 | = 4 , | E 2 | = 7) ( E 2 is formed from all possible paths of players in Π 1 ) 22
Proof Sketch: PoA ≤ 2( ℓ + log n ) E 0 E 1 . . . E s − 1 E s Π 0 Π 1 Π s − 1 . . . s ≤ log n (Each step doubles the size of E i .) Max. # times Min. # times edges in edges used by E s − 1 used (only packets packets in Π s − 1 in Π s − 1 use edges in E s − 1 ) | Π s − 1 | · ℓ ≥ ( C − ( s − 1)) · | E s − 1 | C PoA = C opt ≤ 2 ℓ + s − 1. C opt ≥ | Π s − 1 | ≥ | Π s − 1 | | E s | 2 | E s − 1 | Optimal C Every packet in Π s − 1 | E s | ≤ 2 | E s − 1 | must use at least one edge in E s 23
Proof Sketch: κ e − 1 ≤ PoA ≤ c ( κ 2 e + log n ) C = κ e − 1 C = 1 Optimal Nash-routing Worst Case Nash-routing (Players use shortest paths) (Players use longest paths) C = 1 C = n − 1 = κ e − 1 If network is not a cycle, use the largest cycle in the network. 24
Proof Sketch: κ e − 1 ≤ PoA ≤ c ( κ 2 e + log n ) √ 2 ℓ − 3 Combinatorial Lemma If G is 2-connected, then κ e ( G ) ≥ 2 . 2-connected Networks : ℓ = O ( κ 2 e ), so e + log 2 n ). ⇒ PoA = O ( κ 2 PoA ≤ 2( ℓ + log n ) = General Networks : Step 1: Decompose G : tree of 2-conected and acyclic components. Step 2: Many players satisfied in some 2-connected component; Step 3: Extend PoA ≤ 2( ℓ + log n ) to Partial Nash-routing . Step 4: Use 2-connected and Partial Nash-routing results. 25
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