Bottleneck Routing Games on Grids Costas Busch Rajgopal Kannan - PowerPoint PPT Presentation

Bottleneck Routing Games on Grids Costas Busch Rajgopal Kannan Alfred Samman Department of Computer Science Louisiana State University 1

Talk Outline Introduction Basic Game Channel Game Extensions 2

2-d Grid: n n  n nodes n Used in: • Multiprocessor architectures • Wireless mesh networks • can be extended to d-dimensions 3

Each player corresponds to a pair of source-destination Edge Congestion ( 1  ) 3 C e 2  ( ) 2 C e Bottleneck Congestion:   max ( ) 3 C C e  e E 4

A player may selfishly choose an alternative i path with better congestion Player Congestion  3 C i   1 C i     1 3 C C i i Player Congestion: Maximum edge congestion along its path 5

Routing is a collection of paths, p one path for each player Utility function for player : i congestion pc ( p ) C  i i of selected path Social cost for routing : p SC ( p ) C  bottleneck congestion 6

We are interested in Nash Equilibriums p where every player is locally optimal Metrics of equilibrium quality: Price of Stability Price of Anarchy ( ) SC p ( ) SC p min max * ( ) * SC p p ( ) SC p p is optimal coordinated routing * p with smallest social cost

Bends : number of dimension changes  plus source and destination   6 8

Basic congestion games on grids Price of Stability: ( 1 ) O  Price of Anarchy: ( n ) even with constant bends   ( 1 ) O 9

Better bounds with bends Channel games: Path segments are separated according to length range    log Price of anarchy: O n  Optimal solution uses at most bends 10

There is a (non-game) routing algorithm   with bends and   O log n   approximation ratio O log n Optimal solution uses arbitrary number of bends   Final price of anarchy: 3 log O n 11

Solution without channels: Split Games channels are implemented implicitly in space Similar poly-log price of anarchy bounds 12

Some related work: Price of Anarchy Definition Koutsoupias , Papadimitriou [STACS’99]   Price of Anarchy 1 O for sum of congestion utilities [JACM’02] Arbitrary Bottleneck games [INFOCOM’06], [TCS’09]:   Price of Anarchy | E | O NP-hardness 13

Talk Outline Introduction Basic Game Channel Game Extensions 14

Stability is proven through a potential function defined over routing vectors: M ( p ) [ m , m , , m , , m ]    1 2 k N number of players with congestion C i  k 15

In best response dynamics a player move improves lexicographically the routing vector Player Congestion  3 C i   1 C i  [ 2 , 2 , 0 , 0 , 0 ,..., 0 ] [ 0 , 1 , 3 , 0 , 0 ,..., 0 ]   ( ) ( ) M p M p 16

Before greedy move   C i  ( ) ( ) M p M p k M ( p ) [ m , , m , , m , m , , m ]     1 N k k k 1   M ( p ) [ m , , m , , m , m , , m ]           1 k k k 1 N   After greedy move      C k k C i i 17

Existence of Nash Equilibriums Greedy moves give lower order routings Eventually a local minimum for every player is reached which is a Nash Equilibrium 18

Price of Stability Lowest order routing : p min • Is a Nash Equilibrium • Achieves optimal social cost  * ( ) ( ) SC p SC p min ( min  ) SC p  Price of Stability 1 * ( ) SC p 19

Price of Anarchy Optimal solution Nash Equilibrium *  C  1 / 2 C n C High!    Price of anarchy: / 2 ( ) n n * C 20

Talk Outline Introduction Basic Game Channel Game Extensions 21

channels log n Channel holds path segments A j 1  of length in range:  j j [ 2 , 2 1 ] [ 8 , 15 ] A 3 [ 4 , 7 ] A 2 [ 2 , 3 ] A 1 A [ 1 , 1 ] Row: 0 22

different  1 C channels e same channel  2 C e Congestion occurs only with path segments in same channel 23

Consider an arbitrary Nash Equilibrium p Path of player i maximum congestion C i in path 24

* In optimal routing : SC ( p * ) C *  p Optimal path of player i must have a special edge with congestion C C 1    i C i Since otherwise: pc ( p *) C 1 C 1 1 C pc ( p )         i i i i 25

 In Nash Equilibrium social cost is: ( ) SC p C C   0 0 : Edges of Congestion E C 0  : Players that use edges E 0 0 26

First expansion C 1  C C 1    0 0  Special Edges in optimal paths of 0 27

First expansion C 1  C C 1      1 1 0 0  : Special Edges of Congestion at least 1 E C 1  : Players that use edges E 1 1 28

Second expansion C 1 C 2   C 2 C  C 1  C 2  C 2      1 1 0 0  Special Edges in optimal paths of 1 29

Second expansion C 2   C 2 C  1  C C 2  C 2 1  C       1 1 0 0 2 2  : Special Edges of Congestion at least 2 E C 2  : Players that use edges E 2 2 30

In a similar way we can define:  : Special Edges of Congestion at least E C j j  : Players that use edges E j j We obtain expansion sequences:  , , , , E E E E 0 1 2 3      , , , , 0 1 2 3 31

Redefine expansion:  : Special Edges of Congestion at least E C j j whi ch are the majority in some channel r  r - 1 and edges are sufficient ly far in r : 2 1  : Players that use edges E j j 32

   | |   | | E     j   | |     E j   | | ( ) C j    1 j *    j aC     | | E      j | | ( ) E C j     1 j *   a C 33

  | | E      j | | ( ) E C j     1 j *   a C If then     * ( log ) C C n  constant k | | | | E k E  1 j j E  2 | | n Contradiction 34

   Therefore: * ( log ) C O C n Price of anarchy: C      ( log ) ( log ) O n O n * C 35

Tightness of Price of Anarchy Nash Equilibrium Optimal solution *       2 ( ) ( ) C n 1 C C      Price of anarchy: 2 ( ) ( ) n * 36 C

Talk Outline Introduction Basic Game Channel Game Extensions 37

Split game A 0 A 1 A 2 A 3 A 0 A 1 A 2 A 3 2 n Price of anarchy: O  ( log ) 38

d-dimensional grid Channel game      log Price of anarchy: O n   d Split game      2 Price of anarchy: 2 log O n   d 39