1 Multiconstrained QoS Routing: Simple Approximations to Hard Problems Guoliang (Larry) Xue Arizona State University Research Supported by ARO and NSF Collaborators: W. Zhang, J. Tang, A. Sen, and K. Thulasiraman

2 Outline/Progress of the Talk  Problem Definitions  Related Works  Simple K-Approximation Algorithms  Faster Approximation Schemes  Conclusions

3 Multi-Constrained QoS Routing  Given a network where each link e has a cost c(e) and a delay d(e), we are interested in finding a source- destination path whose cost is within a given cost tolerance C and whose delay is within a given delay tolerance D.  This problem is NP-hard. There are many heuristic algorithms which have no performance guarantees, and sophisticated approximation schemes which are too complicated for protocol implementation.  We have designed the fastest approximation schemes, as well as very simple hop-by-hop routing algorithms that have good performance guarantees.

4 Multi-Constrained QoS Routing  We study the general problem where there are K QoS parameters, for any constant K≥2.  We are given an undirected graph G(V, E) where each edge e  E is associated with K nonnegative weights  1 (e),  2 (e), …,  K (e). We are also given a source s and destination t, and K positive constants W 1, …, W K.  The multi-constrained QoS routing problem asks for an s—t path p such that  k (p) ≤ W k, for k=1, 2, …, K.  For simplicity, we assume K=2 for the most part of this talk. In this case, we will talk about cost and delay.

5 Illustration of the Problem (C=W 1, D=W 2 ) s x y z (2, 5) (12, 5) (12, 20) (2, 2) (10, 0) K = 2 W 1 = 16, W 2 = 8 The shortest path with regard to the 1 st edge weight is (s, z) (14, 1) The shortest path with regard to the 2 nd edge weight is (s, y, z) Neither of them is a feasible solution ! Path (s, x, y, z) is a feasible path.

6 Outline/Progress of the Talk  Problem Definitions  Related Works  Simple K-Approximation Algorithms  Faster Approximation Schemes  Conclusions

7 Related Works  J.M. Jaffe, Algorithms for finding paths with multiple constraints, Networks,  S. Chen and K. Nahrstedt, On finding multi-constrained paths, IEEE International Conference on Communications,  X. Yuan, Heuristic algorithms for multiconstrained quality of service routing, IEEE/ACM Transactions on Networking,  R. Hassin, Approximation schemes for the restricted shortest path problems, Mathematics of Operations Research,  D.H. Lorenz and D. Raz, A simple efficient approximation scheme for the restricted shortest path problem, Operations Research Letters,  G. Xue, A. Sen, W. Zhang, J. Tang, K. Thulasiraman; Finding a path subject to many additive QoS constraints; IEEE/ACM Transactions on Networking,  G. Xue, W. Zhang, J. Tang, K. Thulasiraman; Polynomial time approximation algorithms for multi-constrained QoS routing; IEEE/ACM Transactions on Networking, 2008.

8 Related Works  G. Xue; Minimum cost QoS multicast and unicast routing in communication networks; IPCCC’2000/IEEE Transactions on Communications,  A. Junttner et al., Lagrange relaxation based method for the QoS routing problems, IEEE INFOCOM,  A. Goel et al., Efficient computation of delay-sensitive routes from one source to all destinations, IEEE INFOCOM,  T. Korkmaz and M. Krunz, A randomized algorithm for finding a path subject to multiple QoS requirements, Computer Networks,  P. Van Mieghem et al., Concepts of exact QoS routing algorithms, IEEE/ACM Transactions on Networking,  F.A. Kuipers et al., A comparison of exact and eps-approximation algorithms for constrained routing, IFIP NETWORKING,  A. Orda and A. Sprintson., Efficient algorithms for computing disjoint QoS paths, IEEE INFOCOM, 2004.

9 Outline/Progress of the Talk  Problem Definitions  Related Works  Simple K-Approximation Algorithms  Faster Approximation Schemes  Conclusions

10 A Simple Idea  The decision problem is to find a path p such that c(p)≤C and d(p)≤D.  The optimization problem is to find a path p such that max {c(p)/C, d(p)/D} is minimized.  Define l(p) = max {c(p)/C, d(p)/D} as a new path length.  The original problem has a feasible solution if and only if there is a path p such that l(p)≤1.  The optimization problem is NP-hard as well.  The Idea: For each link e, define a new link weight w(e) = max{c(e)/C, d(e)/D}.  The shortest path with respect to w(e) can be computed easily, and is within a factor of 2 from the optimal solution.

11 Illustration of the Concepts (C=W 1, D=W 2 ) s x y z (2, 5) (12, 5) (12, 20) (2, 2) (10, 0) K = 2 W 1 = 16, W 2 = 8 The shortest path with regard to the 1 st edge weight is (s, z), l(p)=20/8. (14, 1) The shortest path with regard to the 2 nd edge weight is (s, y, z), l(p)=11/8. Neither of them is a feasible/optimal solution ! The optimal path is (s, x, y, z), l(p)=7/8

12 A Simple 2-Approximation Algorithm s x y z (2, 5) (12, 5) (12, 20) (2, 2) (10, 0) K = 2 W 1 = 16, W 2 = 8 (14, 1) The shortest path with regard to the new edge weight is (s, y, z) whose path length is 11/8. This path has a length that is guaranteed to be within a factor of 2 from the optimal value. In this case, we have 11/8 ≤ 2×7/8. (2/16, 5/8)5/8 14/16 20/8 12/16 2/8 10/16

13 A Better Greedy 2-Approximation Algorithm s x y z (2, 5) (12, 5) (12, 20) (2, 2) (10, 0) K = 2 W 1 = 16, W 2 = 8 (14, 1) The path found by Greedy is (s, x, z) with path length 1 [0,0] [2/16, 5/8] A path from s to x with path weights [2/16, 5/8] is stored at node x. The path length is 5/8 [12/16, 20/8][12/16, 5/8] The path at node x is chosen because it has the minimum path length [4/16, 7/8][16/16, 6/8] The path at node y is chosen because it has the minimum path length among the unmarked nodes [22/16, 5/8] The optimal solution is (s, x, y, z) with path length 7/8

14 Proof of Correctness  K-Approx:  The central idea used in the proof of K-Approx relies on the following simple fact.  Let x be a point in the K-dimensional Euclidean plane. Then ||x||  ≤||x|| 1 ≤K  ||x||   Greedy:  Greedy never violates the upper-bound on path length used in the proof of K-Approx.

15 Numerical Results  Algorithms compared  Greedy  Previously best known K-approximation algorithm  FPTAS for the OMCP problem  K = 3, W = W 1 = W 2 = W 3  Networks  well-known Internet topologies ArpaNet (20 nodes and 32 edges) and ItalianNET (33 nodes, 67 edges)  randomly generated topologies  BRITE [BRITE]  Waxman model [WaxJSAC88], and have the default parameters set by BRITE  the edge weights were uniformly generated in a given range (we used the range [1,10]).  Three scenarios Infeasible W = 5 Tight W = 10 Loose W = 20 [BRITE] BRITE; [WaxJSAC88] B.M. Waxman; Routing of multipoint connections; IEEE Journal on Selected Areas in Communications; Vol. (1988). (ε = 0.1)

16 On ArpaNet Topology The number of better paths: path p1 is better than path p2 if l(p1) < l(p2) For any path p, its relative error is calculated as (l(p) - l(p OMCP ))/ l(p OMCP ), where p OMCP is the path found by OMCP for the source-destination pair.

17 On Large Random Network Topologies Path quality, eps = 0.1, 100 nodes, 390 links. Scalability of the algorithms, eps= x314, 210x474, 140x560, 160x634.

18 Outline/Progress of the Talk  Problem Definitions  Related Works  Simple K-Approximation Algorithm  Faster Approximation Schemes  Conclusions

19 Approximation Scheme for SMCP  We are given an undirected graph G(V, E) where each edge e  E is associated with K nonnegative weights  1 (e),  2 (e), …,  K (e). We are also given a source s and destination t, and K positive constants W 1, …, W K. We want to find an s-t path p s.t max{  k (p)/ W k, 1≤k≤K} is minimized.  In a paper published in TON’2007, we designed an algorithm that can find a (1+  )-approximation in O(m(n/  ) K-1 ) time.  This is the first FPTAS for the general SMCP problem (K  2).

20 Approximation Scheme for SMCP  The idea follows that used by other researchers in this field.  Find an initial pair of lower and upper bounds not too far away from each other.  Use scaling/rounding/approximate testing to refine the bounds to within a constant factor  Compute an (1+  )-approximation.  The difference is that we got a pair of lower and upper bounds with a constant (K) factor in a single step, using our K-Approx. This leads to faster running time.  O(mn(loglogn+1/  ))  O(mn/  ) for K=2.  However, the problem is slightly different from the DCLC problem. None of the constraints is enforced. Motivation for the second TON paper.

21 Faster Approximation Schemes for OMCP  All previous approximation schemes for OMCP are based on  Initial bounds  Scaling and rounding, and approximate testing  Final solution  Hassin rounds to floor. Lorenz and Raz round to floor plus one, and showed its advantage over that of Hassin.  A simple combination of the two techniques leads to an approximation scheme that is better than both.

22 Basic Definitions Again

23 Decision Version of the Basic Problem

24 A Restricted Decision Version (for comparison)

25 Optimization Version of the Problem

26 The DCLC Problem (used as a subproblem)

27 MCPN: non-negative integers (2 weights)

28 MCPP: positive integers (K weights)

29 Solving MCPP in O(mC K-1 ) Time

30 Solving MCPP in O(mC K-1 ) Time

31 Illustration of the idea (acyclic graph)

32 Solving MCPN in O((m+nlogn)C) Time

33 Solving MCPN in O((m+nlogn)C) Time

34 Solving MCPN in O((m+nlogn)C) Time

35 Non-negative rounding and approximate testing

36 Non-negative rounding and approximate testing

37 Positive rounding and approximate testing

38 Positive rounding and approximate testing

39 The Power of Approximate Testing  Assume UB  2(1+  )LB. Set C =sqrt(LB  UB/(1+  )). Run TEST(C,  ).  If TEST(C,  )  YES, then DCLC <C(1+  ). Decrease UB to C(1+  ).  If TEST(C,  )  NO, then DCLC >C. Increase LB to C.  In both cases, UB/LB is reduced to sqrt((1+  )(UB/LB)).  We will have UB≤2(1+  )LB, after loglog(initial UB/LB ratio) iterations.

40 Faster Approximation Scheme for DCLC  Use the technique of Lorenz and Raz to compute LB and UB of DCLC so that LB ≤ DCLC ≤UB≤n  LB. This takes O((m+nlogn)logn)) time: logn shortest path computations.  Set  N to (logn) 2, and apply TEST N to refine LB and UB so that LB≤ DCLC ≤UB≤2(1+ (logn) 2 )  LB. This takes O(mn) time: loglog(n) TEST N, each requires O((m+nlogn)n/  N ) time.  Set  P to 1, and apply TEST P to refine LB and UB so that LB≤ DCLC ≤UB≤2(1+ 1 )  LB. This takes O(mnlogloglogn) time: loglog(logn) TEST P, each requires O(mn/  P ) time.  Solve MCPP with scaling factor  =(n-1)/(LB  ). This takes O(mn/  ) time.  O(mn(loglogn+1/  ))  O(mn(logloglogn+1/  ))

41 Faster Approximation Scheme for DCLC

42 Faster Approximation Scheme for DCLC

43 Faster Approximation Scheme for DCLC  O(mn(loglogn + 1/  )) time [Lorenz and Raz ORL’2001]  O(mn(logloglogn + 1/  )) time.  Conjecture: O(mn/  ) time both necessary and sufficient.

44 Dimension Reduction: OMCP  DCLC

45 (1+  )(K-1)-Approx to OMCP, via DCLC

46 Faster Approximation Scheme for OMCP

47 Faster Approximation Scheme for OMCP  This is essentially O(m(n/  ) K-1 ) time.

48 Faster Heuristic/Scheme for DMCP

49 Faster Heuristic/Scheme for DMCP  O(mn(n/  ) K-1 ) time [Yuan TON’2002]  O(m(H/  ) K-1 ) time.

50 Running Time

51 Running Time

52 Path Weight Ratios

53 Outline/Progress of the Talk  Problem Definitions  Related Works  Simple K-Approximation Algorithm  Faster Approximation Schemes  Conclusions

54 Conclusions  We know how to compute a shortest path. OSPF has been proposed by IETF as an RFC.  We don’t know how to handle two or more QoS constraints with guaranteed performance.  This is the first approach which is both simple and provably good.  It is as simple as computing a shortest path.  The computed path is within a factor of K from optimal.  From the theoretical point of view, we have designed faster FPTAS for several versions of the problem.

55 THANK YOU!