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