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Optimal 3-terminal cuts and linear programming

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Abstract

Given an undirected graph G=(V,E) and three specified terminal nodes t 1,t 2,t 3, a 3-cut is a subset A of E such that no two terminals are in the same component of G\A. If a non-negative edge weight c e is specified for each e∈E, the optimal 3-cut problem is to find a 3-cut of minimum total weight. This problem is -hard, and in fact, is max--hard. An approximation algorithm having performance guarantee has recently been given by Călinescu, Karloff, and Rabani. It is based on a certain linear-programming relaxation, for which it is shown that the optimal 3-cut has weight at most times the optimal LP value. It is proved here that can be improved to , and that this is best possible. As a consequence, we obtain an approximation algorithm for the optimal 3-cut problem having performance guarantee . In addition, we show that is best possible for this algorithm.

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References

  1. Calinescu, G., Karloff, H., Rabani, Y.: An improved approximation algorithm for MULTIWAY CUT. Proc. ACM Symp. Theory Comput. 1998, pp. 48–52

  2. Calinescu, G., Karloff, H., Rabani, Y.: An improved approximation algorithm for MULTIWAY CUT. J. Comput. Syst. Sci. 60, 564–574 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cheung, K.K.H., Cunningham, W.H., Tang, L.: Optimal 3-terminal cuts and linear programming. Technical Report CORR 2002-24, Dept. of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, August 2002

  4. Chopra, S., Rao, M.R.: On the multiway cut polyhedron. Networks 21, 51–89 (1991)

    MATH  MathSciNet  Google Scholar 

  5. Cunningham, W.H.: The optimal multiterminal cut problem. In: C. Monma, F. Hwang (eds.), Reliability of Computer and Communications Networks, American Math. Soc., 1991, pp. 105–120

  6. Cunningham, W.H., Tang, L.: Optimal 3-terminal cuts and linear programming, Integer programming and combinatorial optimization. Lecture Notes in Comput. Sci. 1610, 114–125 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dahlhaus, E., Johnson, D., Papadimitriou, C., Seymour, P., Yannakakis, M.: The Complexity of multiway cuts, extended abstract, 1983

  8. Dahlhaus, E., Johnson, D., Papadimitriou, C., Seymour, P., Yannakakis, M.: The Complexity of multiterminal cuts. SIAM J. Comput. 23, 864–894 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  9. Freund, A., Karloff, H.: A lower bound of on the integrality ratio of the Calinescu-Karloff-Rabani relaxation for multiway cut. Inf. Process. Lett. 75, 43–50 (2000)

    Article  MathSciNet  Google Scholar 

  10. Karger, D., Klein, P., Stein, C., Thorup, M., Young, N.: Rounding algorithms for a geometric embedding of minimum multiway cut. Proc. ACM Symp. Theory Comput. 1999, pp. 668–678

  11. Karger, D., Klein, P., Stein, C., Thorup, M., Young, N.: Rounding algorithms for a geometric embedding of minimum multiway cut. Math. Oper. Res. 29, 436–461 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON, Canada, K1S 5B6

    Kevin K. H. Cheung

  2. Department of Combinatorics & Optimization, Faculty of Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ON, Canada, N2L 3G1

    William H. Cunningham

  3. Hong Kong Monetary Authority, 55/F, Two International Finance Centre, 8 Finance Street, Central, Hong Kong

    Lawrence Tang

Authors
  1. Kevin K. H. Cheung
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  2. William H. Cunningham
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  3. Lawrence Tang
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Correspondence to Kevin K. H. Cheung.

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Research of this author was supported by NSERC PGSB.

Research supported by a grant from NSERC of Canada.

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Cheung, K., Cunningham, W. & Tang, L. Optimal 3-terminal cuts and linear programming. Math. Program. 106, 1–23 (2006). https://doi.org/10.1007/s10107-005-0668-2

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  • DOI: https://doi.org/10.1007/s10107-005-0668-2

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