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Bottleneck Routing Games on Grids. Costas Busch Rajgopal Kannan Alfred Samman Department of Computer Science Louisiana State University. Talk Outline. Introduction. Basic Game. Channel Game. Extensions. 2-d Grid: . nodes. Used in: Multiprocessor architectures
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Bottleneck Routing Games on Grids Costas Busch RajgopalKannan Alfred Samman Department of Computer Science Louisiana State University
Talk Outline Introduction Basic Game Channel Game Extensions
2-d Grid: nodes • Used in: • Multiprocessor architectures • Wireless mesh networks • can be extended to d-dimensions
Each player corresponds to a pair of source-destination Edge Congestion Bottleneck Congestion:
A player may selfishly choose an alternative path with better congestion Player Congestion Player Congestion: Maximum edge congestion along its path
Routing is a collection of paths, one path for each player Utility function for player : congestion of selected path Social cost for routing : bottleneck congestion
We are interested in Nash Equilibriums where every player is locally optimal Metrics of equilibrium quality: Price of Stability Price of Anarchy is optimal coordinated routing with smallest social cost
number of dimension changes plus source and destination Bends :
Basic congestion games on grids Price of Stability: Price of Anarchy: even with constant bends
Better bounds with bends Channel games: Path segments are separated according to length range Price of anarchy: Optimal solution uses at most bends
There is a (non-game) routing algorithm with bends and approximation ratio Optimal solution uses arbitrary number of bends Final price of anarchy:
Solution without channels: Split Games channels are implemented implicitly in space Similar poly-log price of anarchy bounds
Some related work: Price of Anarchy Definition Koutsoupias, Papadimitriou [STACS’99] Price of Anarchy for sum of congestion utilities [JACM’02] Arbitrary Bottleneck games [INFOCOM’06], [TCS’09]: Price of Anarchy NP-hardness
Talk Outline Introduction Basic Game Channel Game Extensions
Stability is proven through a potential function defined over routing vectors: number of players with congestion
In best response dynamics a player move improves lexicographically the routing vector Player Congestion
Before greedy move After greedy move
Existence of Nash Equilibriums Greedy moves give lower order routings Eventually a local minimum for every player is reached which is a Nash Equilibrium
Price of Stability Lowest order routing : • Is a Nash Equilibrium • Achieves optimal social cost
Price of Anarchy Optimal solution Nash Equilibrium High! Price of anarchy:
Talk Outline Introduction Basic Game Channel Game Extensions
channels Channel holds path segments of length in range: Row:
different channels same channel Congestion occurs only with path segments in same channel
Consider an arbitrary Nash Equilibrium Path of player maximum congestion in path
In optimal routing : Optimal path of player must have a special edge with congestion Since otherwise:
First expansion Special Edges in optimal paths of
Second expansion Special Edges in optimal paths of
In a similar way we can define: We obtain expansion sequences:
If then constant k Contradiction
Therefore: Price of anarchy:
Tightness of Price of Anarchy Nash Equilibrium Optimal solution Price of anarchy:
Talk Outline Introduction Basic Game Channel Game Extensions
Split game Price of anarchy:
d-dimensional grid Channel game Price of anarchy: Split game Price of anarchy: