1. Shadow Prices vs. Vickrey Prices in Multipath Routing Parthasarathy Ramanujam, Zongpeng Li and Lisa Higham University of Calgary Presented by Ajay Gopinathan

2. 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

3. 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

4. DEFINITIONS Shadow prices vs. Vickrey prices 4

5. 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

6. 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

7. Min-cost unicast LP 7

8. 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

9. 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

10. UNDERLYING CONNECTIONS Shadow prices vs. Vickrey prices 10

11. Shadow prices vs. Vickrey prices Proof using linear programming duality Applies to Unicast Multicast Multi-session multicast, multi-session unicast Theorem 1 Shadow prices provide a lower bound on Vickrey prices 11

12. 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 12

13. 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 13 Main Theorem Max shadow price = unit Vickrey price Min shadow price = unit Vickrey gain unit Vickrey gain shadow price unit Vickrey price

14. Shadow prices vs. Vickrey prices 14 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

15. EFFICIENT COMPUTATION Shadow prices vs. Vickrey prices 15

16. 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

17. We design an algorithm for simultaneously computing unit Vickrey prices for all edges for unicast 17 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?

18. Algorithm illustrated 18

19. Algorithm illustrated 19

20. Algorithm illustrated – Step 1 Compute min-cost flow 20

21. Algorithm illustrated – Step 2 Compute residual network 21

22. Algorithm illustrated – Step 2 Compute residual network 22

23. Algorithm illustrated – Step 3 Run all-pair shortest path algorithm on residual network 23

24. Algorithm illustrated – Step 4 For all unsaturated edges in : Output unit Vickrey price = 0 24

25. Algorithm illustrated – Step 4 Otherwise output unit Vickrey price of 25

26. Algorithm illustrated – Step 4 Otherwise output unit Vickrey price of 26

27. Algorithm complexity Min-cost flow All-pair shortest path Overall complexity Naïve algorithm Best known algorithms today Reduced complexity by factor of 27

28. 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