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Multiconstrained QoS Routing: Simple Approximations to Hard Problems

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. Outline/Progress of the Talk. Problem Definitions Related Works

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Multiconstrained QoS Routing: Simple Approximations to Hard Problems

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  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 algorithmsthat 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 W1, …, WK. • The multi-constrained QoS routing problem asks for an s—t path p such that k(p) ≤ Wk, 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=W1, D=W2) (2, 5) s x (12, 20) K = 2 W1 = 16, W2 = 8 (12, 5) (14, 1) (2, 2) y z (10, 0) The shortest path with regard to the 1st edge weight is (s, z) The shortest path with regard to the 2nd 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, 1984. • S. Chen and K. Nahrstedt, On finding multi-constrained paths, IEEE International Conference on Communications, 1998. • X. Yuan, Heuristic algorithms for multiconstrained quality of service routing, IEEE/ACM Transactions on Networking, 2002. • R. Hassin, Approximation schemes for the restricted shortest path problems, Mathematics of Operations Research, 1992. • D.H. Lorenz and D. Raz, A simple efficient approximation scheme for the restricted shortest path problem, Operations Research Letters, 2001. • G. Xue, A. Sen, W. Zhang, J. Tang, K. Thulasiraman; Finding a path subject to many additive QoS constraints; IEEE/ACM Transactions on Networking, 2007. • 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, 2003. • A. Junttner et al., Lagrange relaxation based method for the QoS routing problems, IEEE INFOCOM, 2001. • A. Goel et al., Efficient computation of delay-sensitive routes from one source to all destinations, IEEE INFOCOM, 2001. • T. Korkmaz and M. Krunz, A randomized algorithm for finding a path subject to multiple QoS requirements, Computer Networks, 2001. • P. Van Mieghem et al., Concepts of exact QoS routing algorithms, IEEE/ACM Transactions on Networking, 2004. • F.A. Kuipers et al., A comparison of exact and eps-approximation algorithms for constrained routing, IFIP NETWORKING, 2006. • 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 weightw(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=W1, D=W2) (2, 5) s x (12, 20) K = 2 W1 = 16, W2 = 8 (12, 5) (14, 1) (2, 2) y z (10, 0) The shortest path with regard to the 1st edge weight is (s, z), l(p)=20/8. The shortest path with regard to the 2nd 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 (2, 5) (2/16, 5/8) 5/8 s x (12, 20) 20/8 K = 2 W1 = 16, W2 = 8 (12, 5) 12/16 (14, 1) 14/16 2/8 (2, 2) y z (10, 0) 10/16 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.

  13. A Better Greedy 2-Approximation Algorithm A path from s to x with path weights [2/16, 5/8] is stored at node x. The path length is 5/8 The path at node x is chosen because it has the minimum path length [0,0] [2/16, 5/8] (2, 5) s x K = 2 W1 = 16, W2 = 8 (12, 20) (12, 5) (14, 1) (2, 2) The optimal solution is (s, x, y, z) with path length 7/8 y z (10, 0) [12/16, 5/8] [22/16, 5/8] [12/16, 20/8] [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 The path found by Greedy is (s, x, z) with path length 1

  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 = W1 = W2 = W3 • 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 (ε = 0.1) [BRITE] BRITE; http://www.cs.bu.edu/brite/. [WaxJSAC88] B.M. Waxman; Routing of multipoint connections; IEEE Journal on Selected Areas in Communications; Vol. (1988).

  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(pOMCP))/ l(pOMCP) , where pOMCP is the path found by OMCP for the source-destination pair.

  17. On Large Random Network Topologies Scalability of the algorithms, eps=0.5. 80x314, 210x474, 140x560, 160x634. Path quality, eps = 0.1, 100 nodes, 390 links.

  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 W1, …, WK. We want to find an s-t path p s.t max{k(p)/ Wk, 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(mCK-1) Time

  30. Solving MCPP in O(mCK-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 roundingandapproximate testing

  36. Non-negative roundingandapproximate testing

  37. Positive roundingandapproximate testing

  38. Positive roundingandapproximate 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 TESTN to refine LB and UB so that LB≤DCLC≤UB≤2(1+(logn)2)LB. This takes O(mn) time: loglog(n) TESTN, each requires O((m+nlogn)n/N) time. • Set P to 1, and apply TESTP to refine LB and UB so that LB≤DCLC≤UB≤2(1+1)LB. This takes O(mnlogloglogn) time: loglog(logn) TESTP, 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

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