Atomic Routing Games on Maximum Congestion

Transcription

1 Atomic Routing Games on Maximum Congestion Costas Busch, Malik Magdon-Ismail June 20, 2006.

2 Outline Motivation and Problem Set Up; Related Work and Our Contributions; Proof Sketches; Wrap Up. 1

3 Routing Routing: coustruct good paths given sources and destinations. Communication Networks eg. Internet. Ad-hoc Networks eg. sensor networks. Parallel Architectures eg. Mesh.... 2

4 Routing Routing: coustruct good paths given sources and destinations. Communication Networks eg. Internet. Ad-hoc Networks eg. sensor networks. Parallel Architectures eg. Mesh.... 3

5 Motivation BILL BILL BILL ALICE ALICE ALICE Which path? 4

6 Motivation BILL BILL BILL ALICE ALICE ALICE delay = 15sec 5

7 Motivation BILL BILL BILL ALICE ALICE ALICE delay = 10sec 6

8 Motivation BILL ALICE BILL ALICE BILL ALICE delay = 10sec 7

9 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

10 Routing Games delay = 12sec delay = 15sec delay = 5sec delay = 13sec 4 Player cost pc i : delay of player i s packet. Social cost SC: maximum delay over all players. SC = 15sec Players minimize their player cost selfishly Ideally, social cost should be minimized. 9

11 Quantifying Delay Congestion: C = max i C i = 3 C 4 = 3 1 C 2 = C 3 = 2 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) Player Cost: delay i = Õ(C i + p i ) [LMR95] [BS95] 10

12 Congested Networks C D, C i p i Social Cost: max i delay i = O(C) Player Cost: delay i = Õ(C i ) 11

13 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 pi 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

14 Quality of Nash-Routings Prics of Stability PoS = inf p P SC(p) SC, Price of Anarchy PoA = sup p P SC(p) SC. PoS: minimum price for stability. (best possible selfish outcome) PoA: maximum price for stability. (worst possible selfish outcome) Ideal: PoS = PoA = 1. 13

15 Related Work Atomic Flow Splittable Flow,,[BM06] Pure 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

16 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

17 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). Theorem 3 κ e 1 PoA c(κ 2 e + log 2 n). κ e (G) is the length of the longest cycle. PoA is bounded by topological properties of the network. 16

18 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

19 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

20 Proof Sketch: PoA 2(l + log n) C 1 C 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. ( E0 : Edges of congestion C. Π 0 : Players using edges in E 0. ) 19

21 Proof Sketch: PoA 2(l + log n) C 1 C C 1 C 1 Π 1 Π 0 Π 1 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

22 Proof Sketch: PoA 2(l + log n) C 2 C C 1 C 2 C 1 C 2 C 1 Π 1 Π 0 Π 1 Alternative paths for players in Π 1 must all have at least one edge with congestion at least C 2. ( E1 : Edges of congestion at least C 1. Π 1 : Players using edges in E 1. ) 21

23 Proof Sketch: PoA 2(l + log n) C 2 C C 1 C 2 C 1 C 2 C 1 Π 1 Π 0 Π 1 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

24 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 edges used by packets in Π s 1 Min. # times edges in E s 1 used (only packets in Π s 1 use edges in E s 1 ) Π s 1 l (C (s 1)) E s 1 C opt Π s 1 E s Π s 1 2 E s 1 PoA = C C opt 2l + s 1. Optimal C Every packet in Π s 1 must use at least one edge in E s E s 2 E s 1 23

25 Proof Sketch: κ e 1 PoA c(κ 2 e + log n) C = 1 C = κ e 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

26 Proof Sketch: κ e 1 PoA c(κ 2 e + log n) Combinatorial Lemma If G is 2-connected, then κ e (G) 2l connected Networks: l = O(κ 2 e ), so PoA 2(l + log n) = PoA = O(κ 2 e + log 2 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 P oa 2(l + log n) to Partial Nash-routing. Step 4: Use 2-connected and Partial Nash-routing results. 25

27 Wrap Up Studied general congestion games with max. social/player costs. Appropriate metrics when delays are important in congested networks. PoS = 1 and selfish dynamics are good. Path Length Bound on PoA: PoA 2(l + log n). Topological bounds on PoA: κ e 1 PoA c(κ 2 e + log 2 n). Conjecture[Lower bound is tight]: PoA κ e. Non-congested networks: SC = C + D; pc i = C i + p i? Thank You! magdon 26