1. Fast Matching Algorithms for Repetitive Optimization Sanjay Shakkottai, UT Austin Joint work with Supratim Deb (Bell Labs) and Devavrat Shah (MIT)

2. Outline Refresher: MWM Background: Switch Scheduling Algorithm and Main Result Unsolved Problems, Extensions, Other Applications Conclusions

3. Maximum Weight Matching in a Bipartite Graph Weight for each edge Weight of matching =Sum of weights of matched edges MWM maximizes the weight of the matching Popular algorithms for obtaining MWM are O(N 3 )

4. Scheduling in Input Buffered Switches Slotted System Slot Duration=Packet transfer time At each slot, an input port can deliver packet to at most one output An output port can receive packet from one input port The schedule corresponds to a matching

5. Popular Scheduling Schemes iSLIP (used in Cisco Routers) Low complexity and High Delay Batch Scheduling Apply MWM once every L slots Does not provide good tradeoff between delay and complexity MWM based on queue-lengths High complexity and low delay

6. Why MWM? Excellent Delay Properties Comparable to output- buffered switches Total queue-length grows linearly with switch size Provides 100% throughput

7. Goal Can we improve the complexity of MWM? Use matching from the previous slot Queue-lengths do not change by much in successive slots

8. Model and Notations An arrival happens at an input port i and destined to output port k in a slot with probability ik Stability if and only if q ik (t) is the number of packets at input port i, destined for output port k at time t

9. Primal and the Dual Problem Primal: Max Subject to Dual: Min Subject to Facts: 1. Can ignore the integrality constraint 2. There exists integral dual solutions

10. Key Idea (x,r,p) optimal if x ij =1 ) d ij =r i +p j -q ij =0 (complementary slackness CS) (x,r,p) feasible (F) As the q ij ‘s change by +1 or –1, adjust the r’s and p’s by adding +1 and –1 so that CS and F are maintained

11. Basic Algorithm Suppose q 11 increases by +1 If d 11 >0, CS and F not violated

12. Basic Algorithm Suppose q 11 increases by +1 If d 11 >0, CS and F not violated If d 11 =0, add +1 to r’s and subtract –1 from p’s till CS and F are satisfied

13. Basic Algorithm Suppose q 11 increases by +1 If d 11 >0, CS and F not violated If d 11 =0, add +1 to r’s and subtract –1 from p’s till CS and F are satisfied

14. Basic Algorithm Suppose q 11 increases by +1 If d 11 >0, CS and F not violated If d 11 =0, add +1 to r’s and subtract –1 from p’s till CS and F are satisfied

15. Basic Algorithm Suppose q 11 increases by +1 If d 11 >0, CS and F not violated If d 11 =0, add +1 to r’s and subtract –1 from p’s till CS and F are satisfied

16. Basic Algorithm Suppose q 11 increases by +1 If d 11 >0, CS and F not violated If d 11 =0, add +1 to r’s and subtract –1 from p’s till CS and F are satisfied

17. Basic Algorithm Suppose q 11 increases by +1 If d 11 >0, CS and F not violated If d 11 =0, add +1 to r’s and subtract –1 from p’s till CS and F are satisfied

18. Complexity Run the basic algorithm for each q ij that changes Complexity is O(N 2 + NE) where E=no of non-empty queues Need to take special care of nodes having zero queues

19. Theorem If  < 0.5, given an MWM from the previous slot, a new MWM can be computed in expected O(N 2 ) operations Conjectured to be true for  <1 Require total queue-length to be O(N) under MWM (simulations suggest so) Conjecture: The expected complexity is O(Nlog(N))

20. Extensions and Applications Improve the complexity bound Devise good incremental MWM algorithm for a more general graph