1. 1 Stateless Optimization of Multi-Commodity Flow Baruch Awerbuch JHU Rohit Khandekar IBM Watson TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A AAAA A

2. 2 Main Issue: avoiding congestion Main result: Greedy agents operating without coordination can minimize congestion in poly-logarithmic time

3. 3 Concurrency causes oscillations Best response: least loaded path Because of concurrency: becomes “worst” response Control is needed to avoid oscillations

4. 4 Internet perspective Since 70’s: Load-Sensitive routing discarded Fixed path routing used Routing paths are highly vulnerable to DOS attacks masquerading as congestion

5. 5 Our framework Agents route commodities through a flow- network and share network bandwidths There is a certain Social objective –Min the maximum congestion on the links Agents are greedy –act greedily to minimize their own cost; no regard to social objective Greedy behavior often leads to highly sub- optimal performance or even system collapse

6. 6 Our approach Impose “rules of conduct” on the agents Stateless local rules: easy to enforce locally without any coordination and without keeping track of history Induce agents to concurrently converge to a near-optimum social objective quickly (typically in poly-logarithmic time)

7. 7 Traditional approach: Analyze Nash equilibrium –No agent has an incentive to move unilaterally –Poly-time convergence to Nash via sequential moves –Or, simpler yet, ignore convergence issue all together Does this make sense in a distributed and dynamic system? –System is distributed: agents don’t move sequentially –In poly-time system changes; thus no convergence To Nash …

8. 8 We define a notion of aggregate equilibrium. –Where system state does not change by too much in long- enough period of time Aggregate equilibrium implies near-optimality. While not in aggregate equilibrium: –Irreversible significant progress Eventually in Aggregate equilibrium. … or Not To Nash?

9. 9 Concurrent Multi-commodity Flows a graph G=(V,E,C); edge-capacities c(e) k commodities: –source s i, sink t i, demand d i ≥ 0 For each commodity: route d i flow between s i and t i such that the maximum edge congestion is minimized. f(e) congestion(e)= total flow thru e capacity of e = u(e) ∑ i f i (e) = u(e)

10. 10 Concurrent Multi-commodity Flows c e = capacity

11. 11 Concurrent Multi-commodity Flows d(1) d(5) d(2) d(4) d(3) Route all demands and minimize the max edge-congestion.

12. 12 Previous sequential solution Many “combinatorial” algorithms known Shahrokhi-Matula (1990) Klein-Plotkin-Stein-Tardos (1990) Leighton-Makedon-Plotkin-Stein-Tardos- Tragoudas (1991) Plotkin-Shmoys-Tardos (1991) Garg-Könemann (1998) Fleischer (2000) Young (2001)

13. 13 Previous Work Even-Dar and Mansour 05: complete network –symmetric strategy space Fisher, Räcke, Vöcking 06: another congestion model –Infinitely many agents each controlling infinitesimal flow. –Single commodity (symmetric strategy space). Fisher & Vöcking (2004), Chien & Sinclair (2007): –Sequential games –polynomial convergence to Nash equilibrium

14. 14 Stateless algorithms Algorithms reacting to the current state of the system without keeping history Output = function (State) Greedy algorithms are a special case of stateless algorithms

15. 15 Properties of stateless alg’s Incremental operation: we do not start from scratch upon each change Self-stabilization: system “corrects” itself after transient failures There is no need to initialize consistently

16. 16 Components of our framework Load-sensitive pricing of the edges –flow agents are forces to pay these prices Flow control (speed limit) rule – cannot increase or drop the flow too fast Profit margin (inertia) rule: – rerouting must yield  profit margin

17. 17 Opportunity cost Cost of an edge with flow f = (m 1/ε ) f(e) congestion Opportunity cost

18. 18 Algorithmic Framework We want to minimize the maximum flow through any edge: minimize max e f(e) We use a smooth convex “equivalent” function: minimize ф = ∑ e (m 1/ε ) f(e) Fact: m O(1) -approx. of ф implies (1+O(ε))- approx. of maximum congestion ф f

19. 19 Maintain the correct estimate of the derivative: During the flow rerouting, the lengths l(e) should not change by more than a factor of (1+ε). Δl(e) = l(e) · log (m 1/ε ) · Δf(e) ≤ l(e) · ε Δf(e)≤ log m Concurrent Algorithmic Framework ε2ε2 Flow control constraint

20. 20 Flow control for concurrency A flow can’t increase by more than 1+  + A flow can’t decrease by more than 1-  -  - = L ¢  +, i.e., downwards speed limit is more aggressive than upwards limit Agents are forced to obey the speed limits

21. 21 Effect of speed limit Fast increase, slow decrease time Flow (log scale)

22. 22 Inertia rule Profit margin (inertia) rule: – rerouting must yield  profit margin a c b d 1+  1

23. 23 Algorithm run by each flow Graph; residual capacity = speed limits while –non-saturated path exists at a cost of (1-  below the average cost, and – Less than 1-  -  fraction of demand rerouted Saturate this path, by increasing its flow to 1+  + times the flow on the bottleneck edge Compensate by proportional uniform decrease

24. 24 Blocking Flow along Shortest Paths

25. 25 Blocking Flow along Shortest Paths

26. 26 Blocking Flow along Shortest Paths

27. 27 Blocking Flow along Shortest Paths

28. 28 Summary : Bounded Best Response Dynamics We impose congestion-sensitive (exponential) edge-costs. Each agent reroutes its flow to minimize its own cost subject to –flow control rule: can’t ramp up too fast – inertia rule: don’t bother with minor improvements Does this bounded best response dynamics converge to a near-optimum solution? –If yes, how fast?

29. 29 Main idea of proof We define the notion of aggregate equilibrium (weaker than Nash) We show that aggregate equilibrium yield near-optimality We show that non-equilibrium state will eventually involve large improvement in a potential function

30. 30 Showing potential decrease Without speed limits, it would be easier to claim potential improvement in moving from expensive to cheap routes We show that speed limit achieves the same, in spite of “ghost chasing” problem, namely shortest path changing very frequently.

31. 31 Main Result Starting from an arbitrary flow, the flow converges to a 1+  approximation to the minimum max-congestion in # of rounds upper bounded by Here m = # edges, |P| = # paths C = max j C j /min j C j Self-stabilizing

32. 32 Conclusion These ideas can be extended to other packing and flow problems. Open question: Eliminate the dependency on L in the convergence time and get a completely poly- logarithmic convergence?