Advertisement

Algorithmica

, Volume 59, Issue 1, pp 81–93 | Cite as

Approximation Algorithms for k-hurdle Problems

  • Brian C. Dean
  • Adam Griffis
  • Ojas Parekh
  • Adam Whitley
Article
  • 102 Downloads

Abstract

The polynomial-time solvable k-hurdle problem is a natural generalization of the classical s-t minimum cut problem where we must select a minimum-cost subset S of the edges of a graph such that |pS|≥k for every s-t path p. In this paper, we describe a set of approximation algorithms for “k-hurdle” variants of the NP-hard multiway cut and multicut problems. For the k-hurdle multiway cut problem with r terminals, we give two results, the first being a pseudo-approximation algorithm that outputs a (k−1)-hurdle solution whose cost is at most that of an optimal solution for k hurdles. Secondly, we provide a \(2(1-\frac{1}{r})\)-approximation algorithm based on rounding the solution of a linear program, for which we give a simple randomized half-integrality proof that works for both edge and vertex k-hurdle multiway cuts that generalizes the half-integrality results of Garg et al. for the vertex multiway cut problem. We also describe an approximation-preserving reduction from vertex cover as evidence that it may be difficult to achieve a better approximation ratio than \(2(1-\frac{1}{r})\). For the k-hurdle multicut problem in an n-vertex graph, we provide an algorithm that, for any constant ε>0, outputs a ⌈(1−ε)k⌉-hurdle solution of cost at most O(log n) times that of an optimal k-hurdle solution, and we obtain a 2-approximation algorithm for trees.

Keywords

Multiway cut Multicut Approximation algorithm Randomized rounding 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Avidor, A., Langberg, M.: The multi-multiway cut problem. In: Proceedings of the 9th Scandinavian Workshop on Algorithm Theory (SWAT), pp. 273–284 (2004) Google Scholar
  2. 2.
    Barany, I., Edmonds, J., Wolsey, L.A.: Packing and covering a tree by subtrees. Combinatorica 6(3), 221–233 (1986) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Burch, C., Carr, R., Krumke, S., Marathe, M., Phillips, C.A.: A decomposition-based pseudoapproximation algorithm for network flow inhibition. In: Woodruff, D.L. (ed.) Network Interdiction and Stochastic Integer Programming, pp. 51–68. Kluwer Academic, Dordrecht (2003) CrossRefGoogle Scholar
  4. 4.
    Burch, C., Krumke, S., Marathe, M., Phillips, C.A., Sundberg, E.: Multicriteria approximation through decomposition. Technical Report (1997) Google Scholar
  5. 5.
    Călinescu, G., Karloff, H.J., Rabani, Y.: An improved approximation algorithm for MULTIWAY CUT. J. Comput. Syst. Sci. 60(3), 564–574 (2000) zbMATHCrossRefGoogle Scholar
  6. 6.
    Carr, R., Fujito, T., Konjevod, G., Parekh, O.: A 2 1/10-approximation algorithm for a generalization of the weighted edge-dominating set problem. J. Comb. Optim. 5, 317–326 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cheung, K., Cunningham, W.H., Tang, L.: Optimal 3-terminal cuts and linear programming. Math. Program. 106(1), 1–23 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cunningham, W.H.: Optimal attack and reinforcement of a network. J. ACM 32(3), 549–561 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    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) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton (1962) zbMATHGoogle Scholar
  11. 11.
    Fulkerson, D.R., Harding, G.C.: Maximizing the minimum source-sink path subject to a budget constraint. Math. Program. 13, 116–118 (1977) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Garg, N., Vazirani, V.V., Yannakakis, M.: Approximate max-flow min-(multi)cut theorems and their applications. SIAM J. Comput. 25(2), 235–251 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Garg, N., Vazirani, V.V., Yannakakis, M.: Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18(1), 3–20 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Garg, N., Vazirani, V.V., Yannakakis, M.: Multiway cuts in node weighted graphs. J. Algorithms 50(1), 49–61 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Gaur, D., Ibaraki, T., Krishnamurti, R.: Constant ratio approximation algorithms for the rectangle stabbing problem and the rectilinear partioning problem. J. Algorithms 43, 138–152 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Goldberg, A., Tarjan, R.: Finding minimum-cost circulations by successive approximation. Math. Oper. Res. 15, 430–466 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Golovin, D., Nagarajan, V., Singh, M.: Approximation the k-multicut problem. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 621–630 (2006) Google Scholar
  18. 18.
    Israeli, E., Wood, R.K.: Shortest path network interdiction. Networks 40(2), 97–111 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Karger, D.R., Klein, P.N., Stein, C., Thorup, M., Young, N.E.: Rounding algorithms for a geometric embedding of minimum multiway cut. Math. Oper. Res. 29(3), 436–461 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Krumke, S., Noltemeier, H., Ravi, R., Schwarz, S., Wirth, H.-C.: Flow improvement and flows with fixed costs. In: Proceedings of the International Conference on Operations Research (OR), pp. 158–167 (1998) Google Scholar
  21. 21.
    Levin, A., Segev, D.: Partial multicuts in trees. In: Proceedings of the 3rd International Workshop on Approximation and Online Algorithms (WAOA), pp. 320–333 (2005) Google Scholar
  22. 22.
    Schwarz, S., Krumke, S.O.: On budget-constrained flow improvement. Inf. Process. Lett. 66(3), 291–297 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Nishihara, O., Inoue, K.: An algorithm for a multiple disconnecting set problem. Unpublished manuscript, Department of Aeronautical Engineering, Kyoto University (1988) Google Scholar
  24. 24.
    Nishihara, O., Kumamoto, H., Inoue, K.: An algorithm for a multiple cut problem and its application. In: 13th International Symposium on Mathematical Programming (1988) Google Scholar
  25. 25.
    Phillips, C.A., Swiler, L.P.: A graph-based system for network-vulnerability analysis. In: Proceedings of the DARPA Information Survivability Conference and Exposition, pp. 71–79 (2000) Google Scholar
  26. 26.
    Phillips, C.A.: The network inhibition problem. In: Proceedings of the 25th Annual ACM Symposium on Theory of Computing (STOC), pp. 776–785 (1993) Google Scholar
  27. 27.
    Wagner, D.K.: Disjoint (s,t)-cuts in a network. Networks 20, 361–371 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Wagner, D.K., Wan, H.: A polynomial-time simplex method for the maximum k-flow problem. Math. Program. 30, 115–123 (1993) CrossRefMathSciNetGoogle Scholar
  29. 29.
    Wood, R.K.: Deterministic network interdiction. Math. Comput. Model. 17(2), 1–18 (1993) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Brian C. Dean
    • 1
  • Adam Griffis
    • 1
  • Ojas Parekh
    • 2
  • Adam Whitley
    • 1
  1. 1.School of ComputingClemson UniversityClemsonUSA
  2. 2.Mathematics and Computer Science DepartmentEmory UniversityAtlantaUSA

Personalised recommendations