1. Connection Preemption in Multi-Class Networks Fahad Rafique Dogar Carnegie Mellon University, USA Collaborators: Laeeq Aslam and Zartash Uzmi (LUMS, Pakistan) Sarmad Abbasi (NUCES, Pakistan) Young-Chon Kim (Chonbuk National University, Korea)

2. 2 Agenda Preemption Problem Earlier Work Our Contribution Conclusion

3. 3 Problem Scenario 1. New connection request (R1,R8,bw,class) 2. Makes an admission control decision If enough bandwidth is available then accept the request; otherwise reject the request A third possibility: accept the request by preempting lower priority connections 5. Preemption decision for R6->R7 6. Preemption decision for R7->R4 7. Preemption decision for R4->R8 We consider the problem of which connections to preempt!!! 3. Makes a constraint based routing decision Say route ={R1->R6->R7->R4->R8} 4.Makes a preemption decision for R1->R6

4. 4 Preemption Problem: Constraint and Objectives What is the constraint while making the preemption decision?  available bw + preempted bw bw of new request Some possible objectives? 1.Minimize the number of preempted connections 2.Minimize the preempted bandwidth 3.Minimize the priority of preempted connections We consider 1 and 2, in that order

5. 5 Earlier Work MinnConn [Peyravian et al. Infocom99] Enhanced version of our problem  Considers priority as the third objective, so tries to achieve objectives 1,2, and 3, in that order  Let’s assume that priority of preemptable connections is the same i.e., we only consider bronze class traffic for preemption. So MinnConn=Our Problem Authors’ claim: MinnConn solves the problem optimally in polynomial time MinnConn runs in polynomial time but is not optimal

6. 6 Our Contribution We show that solving this problem optimally in polynomial time is highly unlikely  Prove that this problem is NP-complete by reducing it to the subset sum problem Propose exact and approximate algorithms to solve this problem  Exact algorithm is optimal and runs in exponential time  Polynomial time approximation algorithm gives a bounded difference from the optimal

7. 7 NP-completeness Proof Subset Sum (SS) Problem  Given a set V={a 1,…,a n } of n positive integers and a number t, is there any subset S of V, such that How is it different from our problem?  Yes/No problem (rather than finding a set)  Sum is made equal to threshold (rather than overshoot)  No restriction on the cardinality of the solution subset This difference is the key to reducing our problem to the subset sum problem 3 differences

8. 8 Proof (Contd.) How to solve the SS problem using the solution to our problem? Basic idea: Pre-Processing SS Input Our Problem Solver Input to our problem Post-Processing Output of our problem Output of SS

9. 9 Proof (Contd.) SS Input: V={a 1,…,a n } and t We construct V’={c 1,b 1 …,c n,b n } and t’ Ensures that exactly n elements are chosen Ensures that either c i or the corresponding b i (but not both) is selected Ensures that those c i s are chosen that minimize the overshoot from the threshold

10. 10 Proof --- Putting it together SS Input: V={a 1,…,a n } and t Check whether the sum of all elements exceed threshold (if not then no solution subset exists) Construct V’={c i s, b i s} and t’ Our Problem Solver V’ and t’ S’{c i s, b i s} Discard the dummy elements (b i s) from S’ Keep the l most significant bits of c i s If their sum equals threshold then output YES else NO SS Output: YES/NO Polynomial Complexity iff SS problem can be solved in polynomial time Pre-Processing Post-Processing

11. 11 Exact Algorithm (V,t,K) In any iteration i, the length of L can be as long as 2 i

12. 12 Approximate Algo. Similar to the exact algorithm but uses a trim function to reduce the length of L in each iteration Trimming:  If two values are quite close (within some factor (1+ δ)) then we can keep the larger one and discard the smaller value Keeping the larger value ensures that our solution is feasible though not optimal But solution is within (1+ δ) K of the optimal  simulation results show that actual difference is much less

13. 13 Conclusion Our contribution  Proof of NP-Completeness  Exact algorithm  Approximate Algorithm Other applications of this problem  Process preemption in OS  Job preemption in scheduling systems

14. 14 Questions?