Shadow Prices and Vickrey Prices in Multipath Routing: An Analytical Approach

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?

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

4. Shadow prices vs. Vickrey prices Definitions

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

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

7. Min-cost unicast LP

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

9. Unit Vickreyprice/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?

10. Shadow prices vs. Vickrey prices Underlying connections

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

12. Shadow prices vs. Vickrey prices • Similar proof technique Theorem 1 Shadow prices provide a lower bound on Vickrey prices Theorem 2 Shadow prices are upper bounded by unit Vickrey prices

13. 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

14. 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

15. Shadow prices vs. Vickrey prices Efficient computation

16. Computing unit Vickreyprices/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

17. 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 alledges for unicast

18. Algorithm illustrated

19. Algorithm illustrated

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

21. Algorithm illustrated – Step 2 Computeresidual network

22. Algorithm illustrated – Step 2 Computeresidual network

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

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

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

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

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

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 Vickreyprice • Min shadow price = unit Vickrey gain • Efficient computation of unit Vickreyprices