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Unsplittable Flow in Paths and Trees and Column-Restricted Packing Integer Programs

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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2009, RANDOM 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5687))

  • 2010 Accesses

Abstract

We consider the unsplittable flow problem (UFP) and the closely related column-restricted packing integer programs (CPIPs). In UFP we are given an edge-capacitated graph G = (V,E) and k request pairs R 1, …, R k , where each R i consists of a source-destination pair (s i ,t i ), a demand d i and a weight w i . The goal is to find a maximum weight subset of requests that can be routed unsplittably in G. Most previous work on UFP has focused on the no-bottleneck case in which the maximum demand of the requests is at most the smallest edge capacity. Inspired by the recent work of Bansal et al. [3] on UFP on a path without the above assumption, we consider UFP on paths as well as trees. We give a simple O(logn) approximation for UFP on trees when all weights are identical; this yields an O(log2 n) approximation for the weighted case. These are the first non-trivial approximations for UFP on trees. We develop an LP relaxation for UFP on paths that has an integrality gap of O(log2 n); previously there was no relaxation with o(n) gap. We also consider UFP in general graphs and CPIPs without the no-bottleneck assumption and obtain new and useful results.

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References

  1. Andrews, M., Chuzhoy, J., Khanna, S., Zhang, L.: Hardness of the undirected edge-disjoint paths problem with congestion. In: Proc. of IEEE FOCS, pp. 226–241 (2005)

    Google Scholar 

  2. Azar, Y., Regev, O.: Combinatorial algorithms for the unsplittable flow problem. Algorithmica 441(1), 49–66 (2006)

    Article  MathSciNet  Google Scholar 

  3. Bansal, N., Friggstad, Z., Khandekar, R., Salavatipour, M.R.: A logarithmic approximation for unsplittable flow on line graphs. In: Proc. of ACM-SIAM SODA, pp. 702–709 (2009)

    Google Scholar 

  4. Bansal, N., Chakrabarti, A., Epstein, A., Schieber, B.: A Quasi-PTAS for unsplittable flow on line graphs. In: Proc. of ACM STOC, pp. 721–729 (2006)

    Google Scholar 

  5. Bar-Noy, A., Bar-Yehuda, R., Freund, A., Naor, J., Schieber, B.: A unified approach to approximating resource allocation and scheduling. JACM 48(5), 1069–1090 (2001)

    Article  MathSciNet  Google Scholar 

  6. Bar-Noy, A., Guha, S., Naor, J., Schieber, B.: Approximating the Throughput of Multiple Machines in Real-Time Scheduling. SICOMP 31(2), 331–352 (2001)

    MathSciNet  MATH  Google Scholar 

  7. Baveja, A., Srinivasan, A.: Approximation algorithms for disjoint paths and related routing and packing problems. Math. Oper. Res. 25(2), 255–280 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bienstock, D.: Approximate formulations for 0-1 knapsack sets. Oper. Res. Lett. 36(3), 317–320 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Calinescu, G., Chakrabarti, A., Karloff, H., Rabani, Y.: Improved approximation algorithms for resource allocation. In: Proc. of IPCO, pp. 439–456 (2001)

    Google Scholar 

  10. Carr, R.D., Fleischer, L., Leung, V.J., Phillips, C.A.: Strengthening integrality gaps for capacitated network design and covering problems. In: Proc. of ACM-SIAM SODA, pp. 106–115 (2000)

    Google Scholar 

  11. Carr, R., Vempala, S.: Randomized meta-rounding. Random Structures and Algorithms 20(3), 343–352 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chakrabarti, A., Chekuri, C., Gupta, A., Kumar, A.: Approximation Algorithms for the Unsplittable Flow Problem. Algorithmica 47(1), 53–78 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chekuri, C., Khanna, S., Shepherd, F.B.: An \(O(\sqrt{n})\) Approximation and Integrality Gap for Disjoint Paths and Unsplittable Flow. Theory of Computing 2, 137–146 (2006)

    Article  MathSciNet  Google Scholar 

  14. Chekuri, C., Khanna, S., Shepherd, F.B.: Edge-Disjoint Paths in Planar Graphs with Constant Congestion. In: Proc. of ACM STOC, pp. 757–766 (2006)

    Google Scholar 

  15. Chekuri, C., Mydlarz, M., Shepherd, F.B.: Multicommodity Demand Flow in a Tree and Packing Integer Programs. ACM Trans. on Algorithms 3(3) (2007)

    Google Scholar 

  16. Chuzhoy, J., Guruswami, V., Khanna, S., Talwar, K.: Hardness of Routing with Congestion in Directed Graphs. In: Proc. of ACM STOC, pp. 165–178 (2007)

    Google Scholar 

  17. Garg, N., Vazirani, V.V., Yannakakis, M.: Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18(1), 3–20 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Guruswami, V., Khanna, S., Shepherd, B., Rajaraman, R., Yannakakis, M.: Near-Optimal Hardness Results and Approximation Algorithms for Edge-Disjoint Paths and Related Problems. J. of Computer and System Sciences 67(3), 473–496 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Håstad, J.: Clique is Hard to Approximate within n 1ε. Acta Mathematica 182, 105–142 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kleinberg, J.M.: Approximation algorithms for disjoint paths problems. PhD thesis, MIT EECS (1996)

    Google Scholar 

  21. Kolliopoulos, S.G.: Edge-disjoint Paths and Unsplittable Flow. In: Gonzalez, T.F. (ed.) Handbook of Approximation Algorithms and Metaheuristics. Chapman & Hall/CRC, Boca Raton (2007)

    Google Scholar 

  22. Kolliopoulos, S.G., Stein, C.: Approximating disjoint-path problems using greedy algorithms and Packing Integer Programs. Math. Prog. A (99), 63–87 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kolman, P.: A Note on the Greedy Algorithm for the Unsplittable Flow Problem. Information Processing Letters 88(3), 101–105 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kolman, P., Scheideler, C.: Improved bounds for the unsplittable flow problem. J. of Algorithms 61(1), 20–44 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pritchard, D.: Approximability of Sparse Integer Programs. In: Proc. of ESA (to appear, 2009); arXiv.org preprint, arXiv:0904.0859v1, http://arxiv.org/abs/0904.0859

  26. Raghavan, P.: Probabilistic Construction of Deterministic Algorithms: Approximating Packing Integer Programs. JCSS 37(2), 130–143 (1988)

    MathSciNet  MATH  Google Scholar 

  27. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg (2003)

    Google Scholar 

  28. Shepherd, B., Vetta, A.: The Demand Matching Problem. Math. of Operations Research 32(3), 563–578 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sherali, H., Adams, W.P.: A Hierarchy of Relaxations and Convex Hull Characterizations for Mixed-Integer Zero-One Programming Problems. Discrete Applied Mathematics and Combinatorial Operations Research and Computer Science 52 (1994)

    Google Scholar 

  30. Srinivasan, A.: Improved approximations for edge-disjoint paths, unsplittable flow, and related routing problems. In: Proc. of IEEE FOCS, pp. 416–425 (1997)

    Google Scholar 

  31. Srinivasan, A.: New Approaches to Covering and Packing Problems. In: Proc. of ACM-SIAM SODA, pp. 567–576 (2001)

    Google Scholar 

  32. Van Vyve, M., Wolsey, L.A.: Approximate Extended Formulations. Math. Prog. 105, 501–522 (2006)

    Article  MATH  Google Scholar 

  33. Varadarajan, K., Venkataraman, G.: Graph Decomposition and a Greedy Algorithm for Edge-disjoint Paths. In: Proc. of ACM-SIAM SODA, pp. 379–380 (2004)

    Google Scholar 

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Authors and Affiliations

  1. Dept. of Computer Science, University of Illinois, Urbana, IL, 61801, USA

    Chandra Chekuri, Alina Ene & Nitish Korula

Authors
  1. Chandra Chekuri
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  2. Alina Ene
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  3. Nitish Korula
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Editor information

Editors and Affiliations

  1. Dept. of Applied Math and Computer Science, The Weizmann Institute of Science, 76100, Rehovot, Israel

    Irit Dinur

  2. Institute for Computer Science and Applied Mathematics, University of Kiel, Olshausenstr. 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Technion, Computer Science Department, 32000, Haifa, Israel

    Joseph Naor

  4. University of Geneva, Centre Universitaire d’Informatique, Battelle Bat. A, 7 rte de Drize, 1227, Carouge, Switzerland

    José Rolim

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Chekuri, C., Ene, A., Korula, N. (2009). Unsplittable Flow in Paths and Trees and Column-Restricted Packing Integer Programs. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2009 2009. Lecture Notes in Computer Science, vol 5687. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03685-9_4

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  • DOI: https://doi.org/10.1007/978-3-642-03685-9_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03684-2

  • Online ISBN: 978-3-642-03685-9

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