Mathematical Programming

, Volume 102, Issue 1, pp 167–183 | Cite as

Greedy splitting algorithms for approximating multiway partition problems

Article

Abstract.

Given a system (V,T,f,k), where V is a finite set, Open image in new window is a submodular function and k≥2 is an integer, the general multiway partition problem (MPP) asks to find a k-partition Open image in new window ={V1,V2,...,V k } of V that satisfies Open image in new window for all i and minimizes f(V1)+f(V2)+···+f(V k ), where Open image in new window is a k-partition of Open image in new window hold. MPP formulation captures a generalization in submodular systems of many NP-hard problems such as k-way cut, multiterminal cut, target split and their generalizations in hypergraphs. This paper presents a simple and unified framework for developing and analyzing approximation algorithms for various MPPs.

Keywords

Approximation algorithm Hypergraph partition k-way cut Multiterminal cut Multiway partition problem Submodular function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Benczúr, A.A.: Counterexamples for directed and node capacitated cut-trees. SIAM J. Comput. 24, 505–510 (1995)Google Scholar
  2. 2.
    Burlet, M., Goldschmidt, O.: A new and improved algorithm for the 3-cut problem. Oper. Res. Lett. 21, 225–227 (1997)CrossRefGoogle Scholar
  3. 3.
    Calinescu, G., Karloff, H., Rabani, Y.: An improved approximation algorithm for multiway cut. J. Comput. System Sci. 60, 564–574 (2000)CrossRefGoogle Scholar
  4. 4.
    Chopra, S., Owen, J.H.: A note on formulations for the A-partition problem on hypergraphs. Discrete Appl. Math. 90, 115–133 (1999)CrossRefGoogle Scholar
  5. 5.
    Cunningham, W.H.: Optimal attack and reinforcement of a network. J. Assoc. Comput. Mach. 32, 549–561 (1985)Google Scholar
  6. 6.
    Dahlhaus, E., Johnson, D.S., Papadimitriou, C.H., Seymour, P.D, Yannakakis, M.: The complexity of multiterminal cuts. SIAM J. Comput. 23, 864–894 (1994)Google Scholar
  7. 7.
    Gomory, R.E., Hu, T.C.: Multi-terminal network flows. J. Soc. Indust. Appl. Math. 9, 551–570 (1961)Google Scholar
  8. 8.
    Goldschmidt, O., Hochbaum, D.S.: A polynomial algorithm for the k-cut problem for fixed k. Math. Oper. Res. 1,9 24–37 (1994)Google Scholar
  9. 9.
    Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum flow problem. J. Assoc. Comput. Mach. 35, 921–940 (1988)Google Scholar
  10. 10.
    Gusfield, D.: Connectivity and edge-disjoint spanning trees. Inform. Process. Lett. 16, 87–89 (1983)CrossRefGoogle Scholar
  11. 11.
    Garg, N., Vazirani, V.V., Yannakakis, M.: Multiway cuts in directed and node weighted graphs (extended abstract). In Proc. ICALP 1994, LNCS 820, 487–498 (1994)Google Scholar
  12. 12.
    Iwata, S., Fleischer, L.L., Fujishige, S.: A combinatorial strongly polynomial time algorithm for minimizing submodular functions. J. ACM 48, 761–777 (2001)CrossRefGoogle Scholar
  13. 13.
    Kapoor, S.: On minimum 3-cuts and approximating k-cuts using cut trees. In Proc. IPCO 1996, LNCS 1084, 132–146 (1996)Google Scholar
  14. 14.
    Karger, D.R., Klein, P., Stein, C., Thorup, M., Young, N.E.: Rounding algorithms for a geometric embedding of minimum multiway cut. In Proc. STOC 1999, 668–678 (1999)Google Scholar
  15. 15.
    Korte, B., Vygen, J. : Combinatorial Optimization. Theory and Algorithms. Springer, Berlin (2000)Google Scholar
  16. 16.
    Klimmek, R., Wagner, F.: A simple hypergraph min cut algorithm. Technical Report B 96-02, Freie Universität Berlin (1996)Google Scholar
  17. 17.
    Lawler, E.L.: Cutsets and partitions of hypergraphs. Networks 3, 275–285 (1973)Google Scholar
  18. 18.
    Lengauer, T.: Combinatorial Algorithms for Integrated Circuit Layout. Wiley, New York (1990)Google Scholar
  19. 19.
    Lee, C.H., Kim, M., Park, C.I. : An efficient k-way graph partitioning algorithm for task allocation in parallel computing systems. In Proc. IEEE Int. Conf. on Computer-Aided Design 1990, 748–751 (1990)Google Scholar
  20. 20.
    Matula, D.W.: A linear time 2+ε approximation algorithm for edge connectivity. In Proc. SODA 1993, 500–504 (1993)Google Scholar
  21. 21.
    Maeda, N., Nagamochi, H., Ibaraki, T.: Approximate algorithms for multiway objective point split problems of graphs (in Japanese). Computing Devices and Algorithms (in Japanese) (Kyoto, 1993). Surikaisekikenkyusho Kokyuroku 833, 98–109 (1993)Google Scholar
  22. 22.
    Nagamochi, H., Ibaraki, T.: Computing edge connectivity in multigraphs and capacitated graphs. SIAM J. Discrete Math. 5, 54–66 (1992)Google Scholar
  23. 23.
    Narayanan, H., Roy, S., Patkar, S.: Approximation algorithms for min-k-overlap problems using the principal lattice of partitions approach. J. Algorithms 21, 306–330 (1996)CrossRefGoogle Scholar
  24. 24.
    Pulleyblank, W.R.: Presentation at SIAM Meeting on Optimization, MIT, Boston (1992)Google Scholar
  25. 25.
    Queyranne, M.: Minimizing symmetric submodular functions. Math. Program. B 82, 3–12 (1995)CrossRefGoogle Scholar
  26. 26.
    Queyranne, M.: On optimum size-constrained set partitions. AUSSOIS 1999, France (1999)Google Scholar
  27. 27.
    Saran, H., Vazirani, V.V.: Finding k-cuts within twice the optimal. SIAM J. Comput. 24, 101–108 (1995)Google Scholar
  28. 28.
    Tittmann, P.: Partitions and network reliability. Discrete Appl. Math. 95, 445–453 (1999)CrossRefGoogle Scholar
  29. 29.
    Vazirani, V.V.: Approximation Algorithms. Springer-Verlag, Berlin (2001)Google Scholar
  30. 30.
    Zhao, L.: Approximation algorithms for partition and design problems in networks. PhD Thesis, Graduate school of Informatics, Kyoto University, Japan (2002)Google Scholar
  31. 31.
    Zhao, L., Nagamochi, H., Ibaraki, T.: Approximating the minimum k-way cut in a graph via minimum 3-way cuts. J. Comb. Optim. 5, 397–410 (2001)CrossRefGoogle Scholar
  32. 32.
    Zhao, L., Nagamochi, H., Ibaraki, T.: On generalized greedy splitting algorithms for multiway partition problems. Discrete Appl. Math. (to appear). A preliminary version appeared in Proc. ISAAC 2001 (A unified framework for approximating multiway partition problems (extended abstract). LNCS 2223, 682–694) (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Liang Zhao
    • 1
  • Hiroshi Nagamochi
    • 2
  • Toshihide Ibaraki
    • 3
  1. 1.Dept. Information Science, Faculty of EngineeringUtsunomiya UniversityUtsunomiyaJapan
  2. 2.Dept. Information and Computer SciencesToyohashi University of TechnologyToyohashi, AichiJapan
  3. 3.Dept. Applied Mathematics and Physics, Graduate School of InformaticsKyoto UniversityKyotoJapan

Personalised recommendations