Shadow Prices vs.Vickrey Prices
in Multipath Routing
Parthasarathy Ramanujam, Zongpeng Li and Lisa Higham
University of Calgary
Presented by
Ajay Gopinathan
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?
2
Outline

Definitions
◦ Shadow/Vickrey prices in routing

Underlying Connections
◦ Relationship between shadow/Vickrey prices

Efficient Computation
◦ Algorithm for efficient computation of unit
Vickrey prices

Conclusion
3
Shadow prices vs.Vickrey prices
DEFINITIONS
4
Shadow prices

Optimal 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
5
Network model

Communication 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
6
Min-cost unicast LP
7
Vickrey prices

Mechanism design – VCG scheme
◦ Strategyproof mechanism

Network games with selfish agents
◦ Wealth of protocols employing VCG
◦ Requires computation of Vickrey prices

Vickrey price of edge is added cost of
routing when edge is removed
8
Unit Vickrey price/gain

Define unit Vickrey price
◦ Added cost of routing if capacity of edge is
reduced by one
◦ Fine grained version of Vickrey price

Similarly 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?
9
Shadow prices vs.Vickrey prices
UNDERLYING
CONNECTIONS
10
Shadow prices vs.Vickrey prices
Theorem 1
Shadow prices provide a lower bound on Vickrey prices


Proof using linear programming duality
Applies to
◦ Unicast
◦ Multicast
◦ Multi-session multicast, multi-session unicast
11
Shadow prices vs.Vickrey prices
Theorem 1
Shadow prices provide a lower bound on Vickrey prices
Theorem 2
Shadow prices are upper bounded by unit Vickrey prices

Similar proof technique
12
Shadow prices vs.Vickrey prices
Theorem 1
Shadow prices provide a lower bound on Vickrey prices
Theorem 2
Shadow prices are upper bounded by unit Vickrey prices
Main Theorem
Max shadow price = unit Vickrey price
Min 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 price
Min shadow price = unit Vickrey gain
Unit Vickrey gain ≤ Shadow price ≤ Unit Vickrey price

Techniques
◦ Linear programming duality
◦ Negative cycle theorem for min-cost flow
optimality
14
Shadow prices vs.Vickrey prices
EFFICIENT
COMPUTATION
15
Computing unit Vickrey prices/gain

Unit 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
16
Can we do better?
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]
We design an algorithm for simultaneously
computing unit Vickrey prices for all edges
for unicast
17
Algorithm illustrated
18
Algorithm illustrated
19
Algorithm illustrated – Step 1
Compute min-cost flow
20
Algorithm illustrated – Step 2
Compute residual network
21
Algorithm illustrated – Step 2
Compute residual network
22
Algorithm illustrated – Step 3
Run all-pair shortest path algorithm on
residual network
23
Algorithm illustrated – Step 4
For all unsaturated edges in :
Output unit Vickrey price = 0
24
Algorithm illustrated – Step 4
Otherwise output unit Vickrey price of
25
Algorithm illustrated – Step 4
Otherwise output unit Vickrey price of
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Algorithm complexity

Min-cost flow
All-pair shortest path
Overall complexity
Naïve algorithm
Best known algorithms today

Reduced complexity by factor of




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Conclusion
Shadow prices and Vickrey prices
measure importance of a link
 Bounds

◦ 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
28