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Shadow Prices vs. Vickrey Prices in Multipath RoutingParthasarathy Ramanujam, Zongpeng Li and Lisa HighamUniversity of Calgary
Presented byAjay Gopinathan
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Problem Statement
How important is a link for a given information flow in a
network?Known metrics
Shadow prices (optimization)Vickrey prices (economics)
How are shadow prices and Vickrey prices related?
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OutlineDefinitions
◦Shadow/Vickrey prices in routingUnderlying Connections
◦Relationship between shadow/Vickrey prices
Efficient Computation◦Algorithm for efficient computation
of unit Vickrey pricesConclusion
DEFINITIONSShadow prices vs. Vickrey prices
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Shadow pricesOptimal routing can be
formulated as a mathematical program◦Convex, possibly linear
Each constraint => Lagrangian multiplier
Shadow price of constraint is Lagrangian multiplier at optimality◦Dual variables (linear program)
Measure of “importance” of constraint
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Network modelCommunication network model
◦Directed◦Edges have capacity ◦Edges have cost per unit flow◦Source wishes to send data at rate◦Minimize routing costs
Solve using linear programming
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Min-cost unicast LP
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Vickrey pricesMechanism design – VCG scheme
◦Strategyproof mechanismNetwork games with selfish agents
◦Wealth of protocols employing VCG◦Requires computation of Vickrey
pricesVickrey price of edge is added cost
of routing when edge is removed
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Unit Vickrey price/gainDefine unit Vickrey price
◦Added cost of routing if capacity of edge is reduced by one
◦Fine grained version of Vickrey priceSimilarly define unit Vickrey gain
◦Reduced cost of routing if capacity of edge is increased by one
Decision tool for network designer ◦Should link capacity be increased?
UNDERLYING CONNECTIONS
Shadow prices vs. Vickrey prices
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Shadow prices vs. Vickrey prices
Proof using linear programming dualityApplies to
◦Unicast ◦Multicast◦Multi-session multicast, multi-session unicast
Theorem 1 Shadow prices provide a lower bound on Vickrey
prices
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Shadow prices vs. Vickrey prices
Similar proof technique
Theorem 2 Shadow prices are upper bounded by unit
Vickrey prices
Theorem 1 Shadow prices provide a lower bound on Vickrey
prices
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Shadow prices vs. Vickrey prices
Theorem 2 Shadow prices are upper bounded by unit
Vickrey prices
Theorem 1 Shadow prices provide a lower bound on Vickrey
prices
Main Theorem Max shadow price = unit Vickrey priceMin shadow price = unit Vickrey gain
unit Vickrey gain ≤ shadow price ≤ unit Vickrey price
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Shadow prices vs. Vickrey prices
Main Theorem Max shadow price = unit Vickrey priceMin shadow price = unit Vickrey gainUnit Vickrey gain ≤ Shadow price ≤ Unit
Vickrey price
Techniques◦Linear programming duality◦Negative cycle theorem for min-cost
flow optimality
EFFICIENT COMPUTATION
Shadow prices vs. Vickrey prices
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Computing unit Vickrey prices/gainUnit Vickrey prices/gain
◦Importance of upgrading link capacity
Naïve algorithm◦Compute optimal flow cost◦Decrement (increment) edge
capacity by 1◦Compute new flow cost◦Repeat for each edge
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We design an algorithm for simultaneously computing unit Vickrey
prices for all edges for unicast
What is the complexity of computing all Vickrey prices?
[Nisan and Ronen, STOC 1999]
All link Vickrey prices for shortest path
[Hershberger and Suri, FOCS 2001]
Can we do better?
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Algorithm illustrated
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Algorithm illustrated
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Algorithm illustrated – Step 1
Compute min-cost flow
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Algorithm illustrated – Step 2
Compute residual network
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Algorithm illustrated – Step 2
Compute residual network
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Algorithm illustrated – Step 3
Run all-pair shortest path algorithm on residual network
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Algorithm illustrated – Step 4
For all unsaturated edges in : Output unit Vickrey price = 0
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Algorithm illustrated – Step 4
Otherwise output unit Vickrey price of
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Algorithm illustrated – Step 4
Otherwise output unit Vickrey price of
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Algorithm complexityMin-cost flowAll-pair shortest pathOverall complexityNaïve algorithm Best known algorithms today
Reduced complexity by factor of
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ConclusionShadow prices and Vickrey prices
measure importance of a linkBounds
◦Shadow prices ≤ Vickrey prices◦Shadow prices ≤ unit Vickrey prices
Max shadow price = unit Vickrey price◦Min shadow price = unit Vickrey gain
Efficient computation of unit Vickrey prices