Algorithmica

, Volume 68, Issue 3, pp 776–804 | Cite as

Multicommodity Flow in Trees: Packing via Covering and Iterated Relaxation

Article

Abstract

We consider the max-weight integral multicommodity flow problem in trees. In this problem we are given an edge-, arc-, or vertex-capacitated tree and weighted pairs of terminals, and the objective is to find a max-weight integral flow between terminal pairs subject to the capacities. This problem is APX-hard and a 4-approximation for the edge- and arc-capacitated versions is known. Some special cases are exactly solvable in polynomial time, including when the graph is a path or a star.

We show that all three versions of this problems fit in a common framework: first, prove a counting lemma in order to use the iterated LP relaxation method; second, solve a covering problem to reduce the resulting infeasible solution back to feasibility without losing much weight. The result of the framework is a 1+O(1/μ)-approximation algorithm where μ denotes the minimum capacity, for all three versions. A complementary hardness result shows this is asymptotically best possible. For the covering analogue of multicommodity flow, we also show a 1+Θ(1/μ) approximability threshold with a similar framework.

When the tree is a spider (i.e. only one vertex has degree greater than 2), we give a polynomial-time exact algorithm and a polyhedral description of the convex hull of all feasible solutions. This holds more generally for instances we call root-or-radial.

A preliminary version of this work appeared in Könemann et al. (Proc. 6th Int. Workshop Approx. & Online Alg. (WAOA), pp. 1–14, 2008).

Keywords

Multicommodity flow Approximation algorithms Iterated LP relaxation Polyhedral combinatorics 

References

  1. 1.
    Andrews, M., Chuzhoy, J., Guruswami, V., Khanna, S., Talwar, K., Zhang, L.: Inapproximability of edge-disjoint paths and low congestion routing on undirected graphs. Combinatorica 30, 485–520 (2010). doi:10.1007/s00493-010-2455-9 CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Anstee, R.P.: A polynomial algorithm for b-matchings: an alternative approach. Inf. Process. Lett. 24(3), 153–157 (1987). doi:10.1016/0020-0190(87)90178-5 CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bansal, N., Khandekar, R., Nagarajan, V.: Additive guarantees for degree-bounded directed network design. SIAM J. Comput. 39(4), 1413–1431 (2009). doi:10.1137/080734340. Preliminary version In: Proc. 40th Symp. Theory Comp. (STOC), pp. 769–778 (2008) CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Beck, J., Fiala, T.: “Integer-making” theorems. Discrete Appl. Math. 3(1), 1–8 (1981). doi:10.1016/0166-218X(81)90022-6 MATHMathSciNetGoogle Scholar
  5. 5.
    Bonsma, P., Schulz, J., Wiese, A.: A constant factor approximation algorithm for unsplittable flow on paths. In: Proc. 52nd Symp. Found. Comp. Sci. (FOCS), pp. 47–56 (2011). doi:10.1109/FOCS.2011.10 Google Scholar
  6. 6.
    Caprara, A., Fischetti, M.: \(\{0,\frac{1}{2} \}\)-Chvátal–Gomory cuts. Math. Program. 74(3), 221–235 (1996). doi:10.1007/BF02592196 CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Chakrabarty, D., Grant, E., Könemann, J.: On column-restricted and priority covering integer programs. In: Proc. 14th Conf. Int. Prog. Comb. Opt. (IPCO), pp. 355–368 (2010). doi:10.1007/978-3-642-13036-6_27 Google Scholar
  8. 8.
    Chan, T.M., Grant, E., Könemann, J., Sharpe, M.: Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. In: Proc. 23rd Symp. Disc. Alg. (SODA), pp. 1576–1585 (2012). http://dl.acm.org/citation.cfm?id=2095116.2095241 Google Scholar
  9. 9.
    Chekuri, C., Mydlarz, M., Shepherd, F.B.: Multicommodity demand flow in a tree and packing integer programs. ACM Trans. Algorithms (2007). doi:10.1145/1273340.1273343. Preliminary version in Proc. 30th Int. Colloq. Automata, Lang. & Prog. (ICALP), pp. 410–425 (2003) MathSciNetGoogle Scholar
  10. 10.
    Cheriyan, J., Jordán, T., Ravi, R.: On 2-coverings and 2-packings of laminar families. In: Proc. 7th European Symp. Alg. (ESA), pp. 510–520 (1999). doi:10.1007/3-540-48481-7_44 Google Scholar
  11. 11.
    Chlebík, M., Chlebíková, J.: Complexity of approximating bounded variants of optimization problems. Theor. Comput. Sci. 354(3), 320–338 (2006). doi:10.1016/j.tcs.2005.11.029. Preliminary version in Proc. 14th Fund. Comp. Theory (FCT), pp. 27–38 (2003) CrossRefMATHGoogle Scholar
  12. 12.
    Edmonds, J., Johnson, E.: Matching: A well-solved class of integer linear programs. In: Guy, R., Hanani, H., Sauer, N., Schonheim, J. (eds.) Combinatorial Structures and Their Applications (Proc. 1969 Calgary Conf. Comb. Struct. Appl.), pp. 89–92. Gordon and Breach, New York (1970) Google Scholar
  13. 13.
    Erlebach, T.: Approximation algorithms for edge-disjoint paths and unsplittable flow. In: Bampis, E., Jansen, K., Kenyon, C. (eds.) Efficient Approximation and Online Algorithms, pp. 97–134. Springer, Berlin (2006). Chap. 4, http://dl.acm.org/citation.cfm?id=2168139.2168144 CrossRefGoogle Scholar
  14. 14.
    Erlebach, T., Jansen, K.: Conversion of coloring algorithms into maximum weight independent set algorithms. Discrete Appl. Math. 148(1) (2005). doi:10.1016/j.dam.2004.11.007. Preliminary version in Proc. Satellite Workshops 27th ICALP, pp. 135–146 (2000) Google Scholar
  15. 15.
    Erlebach, T., Vukadinović, D.: Path problems in generalized stars, complete graphs, and brick wall graphs. Discrete Appl. Math. 154, 673–683 (2006). doi:10.1016/j.dam.2005.05.017. Preliminary version in Proc. 13th Fund. Comp. Theory (FCT), pp. 483–494 (2001) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Even, S., Itai, A., Shamir, A.: On the complexity of timetable and multicommodity flow problems. SIAM J. Comput. 5(4), 691–703 (1976). doi:10.1137/0205048. Preliminary version in Proc. 16th Symp. Found. Comp. Sci. (FOCS), pp. 184–193 (1975) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Gabow, H.N., Gallagher, S.: Iterated rounding algorithms for the smallest k-edge connected spanning subgraph. SIAM J. Comput. 41(1), 61–103 (2012). doi:10.1137/080732572. Preliminary version in Proc. 19th Symp. Disc. Alg. (SODA), pp. 550–559 (2008) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Gabow, H.N., Goemans, M.X., Tardos, É., Williamson, D.P.: Approximating the smallest k-edge connected spanning subgraph by LP-rounding. Networks 53(4), 345–357 (2009). doi:10.1002/net.20289. Preliminary version in Proc. 16th Symp. Disc. Alg. (SODA), pp. 562–571 (2005) CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Garg, N., Vazirani, V.V., Yannakakis, M.: Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18(1), 3–20 (1997). doi:10.1007/BF02523685. Preliminary version in Proc. 20th Int. Colloq. Automata, Lang. & Prog. (ICALP), pp. 64–75 (1993) CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Goemans, M.X.: Minimum bounded degree spanning trees. In: Proc. 47th Symp. Found. Comp. Sci. (FOCS), pp. 273–282 (2006). doi:10.1109/FOCS.2006.48 Google Scholar
  21. 21.
    Guruswami, V., Khanna, S., Rajaraman, R., Shepherd, F.B., Yannakakis, M.: Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems. J. Comput. Syst. Sci. 67(3), 473–496 (2003). Preliminary version in Proc. 31st Symp. Theory Comp. (STOC), pp. 19–28 (1999) CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Hartman, I.B.-A.: Optimal k-colouring and k-nesting of intervals. In: Proc. 4th Israel Symp. Theory Comput. & Systems, pp. 207–220 (1992) CrossRefGoogle Scholar
  23. 23.
    Jain, K.: A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica 21(1), 39–60 (2001). doi:10.1007/s004930170004. Preliminary version in Proc. 39th Symp. Found. Comp. Sci. (FOCS), pp. 448–457 (1998) CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Kann, V.: On the approximability of NP-complete optimization problems. PhD thesis, Royal Institute of Technology Stockholm (1992) Google Scholar
  25. 25.
    Karp, R.M., Leighton, F.T., Rivest, R.L., Thompson, C.D., Vazirani, U.V., Vazirani, V.V.: Global wire routing in two-dimensional arrays. Algorithmica 2, 113–129 (1987). doi:10.1007/BF01840353 CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Könemann, J., Parekh, O., Pritchard, D.: Max-weight integral multicommodity flow in spiders and high-capacity trees. In: Proc. 6th Int. Workshop Approx. & Online Alg. (WAOA), pp. 1–14 (2008). doi:10.1007/978-3-540-93980-1_1 Google Scholar
  27. 27.
    Lau, L., Naor, J., Salavatipour, M., Singh, M.: Survivable network design with degree or order constraints. SIAM J. Comput. 39(3), 1062–1087 (2009). doi:10.1137/070700620. Preliminary version in Proc. 39th Symp. Theory Comp. (STOC), pp. 651–660 (2007) CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Nagarajan, V., Ravi, R., Singh, M.: Simpler analysis of LP extreme points for traveling salesman and survivable network design problems. Oper. Res. Lett. 38(3), 156–160 (2010). doi:10.1016/j.orl.2010.02.005 CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Nguyen, T.: On the disjoint paths problem. Oper. Res. Lett. 35(1), 10–16 (2007). doi:10.1016/j.orl.2006.02.001 CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Parekh, O.: Iterative packing for demand and hypergraph matching. In: Proc. 15th Conf. Int. Prog. Comb. Opt. (IPCO), pp. 349–361 (2011). doi:10.1007/978-3-642-20807-2_28 Google Scholar
  31. 31.
    Peis, B., Wiese, A.: Throughput maximization for periodic packet routing on trees and grids. In: Proc. 8th WAOA (Workshop Approx. & Online Alg.), pp. 213–224 (2011). doi:10.1007/978-3-642-18318-8_19 CrossRefGoogle Scholar
  32. 32.
    Pritchard, D.: k-edge-connectivity: approximation and LP relaxation. In: Proc. 8th WAOA (Workshop Approx. & Online Alg.), pp. 225–236 (2010). doi:10.1007/978-3-642-18318-8_20 Google Scholar
  33. 33.
    Pritchard, D., Chakrabarty, D.: Approximability of sparse integer programs. Algorithmica 61(1), 75–93 (2011). doi:10.1007/s00453-010-9431-z. Preliminary version in Proc. 17th European Symp. Alg. (ESA), pp. 83–94 (2009) CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Raghavan, P., Thompson, C.: Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica 7, 365–374 (1987). doi:10.1007/BF02579324 CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003). 3-540-44389-4 Google Scholar
  36. 36.
    Shepherd, F.B., Vetta, A.: The demand-matching problem. Math. Oper. Res. 32(3), 563–578 (2007). doi:10.1287/moor.1070.0254. Preliminary version in Proc. 9th Conf. Int. Prog. Comb. Opt. (IPCO), pp. 457–474 (2002) CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Singh, M., Lau, L.C.: Approximating minimum bounded degree spanning trees to within one of optimal. In: Proc. 39th Symp. Theory Comp. (STOC), pp. 661–670 (2007). doi:10.1145/1250790.1250887 Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Jochen Könemann
    • 1
  • Ojas Parekh
    • 2
  • David Pritchard
    • 3
  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  2. 2.Sandia National LaboratoriesAlbuquerqueUSA
  3. 3.Department of Computer SciencePrinceton UniversityPrincetonUSA

Personalised recommendations