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A Fast Algorithm for the Path 2-Packing Problem

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Abstract

Let G be an undirected graph and \(\mathcal{T}=\{T_{1},\ldots,T_{k}\}\) be a collection of disjoint subsets of nodes. Nodes in T 1∪⋅⋅⋅∪T k are called terminals, other nodes are called inner. By a \(\mathcal{T}\) -path we mean a path P such that P connects terminals from distinct sets in \(\mathcal{T}\) and all internal nodes of P are inner. We study the problem of finding a maximum cardinality collection ℘ of \(\mathcal{T}\) -paths such that at most two paths in ℘ pass through any node. Our algorithm is purely combinatorial and has the time complexity O(mn 2), where n and m denote the numbers of nodes and edges in G, respectively.

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References

  1. Babenko, M.: A fast algorithm for path 2-packing problem. Lect. Notes Comput. Sci. 4649, 70–81 (2007)

    Article  Google Scholar 

  2. Babenko, M., Karzanov, A.: Free multiflows in bidirected and skew-symmetric graphs. Discrete Appl. Math. 155(13), 1715–1730 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Babenko, M., Karzanov, A.: A scaling algorithm for the maximum node-capacitated multiflow problem. Lect. Notes Comput. Sci. 5193, 124–135 (2008)

    Article  Google Scholar 

  4. Cherkassky, B.: A solution of a problem on multicommodity flows in a network. Ekon. Mat. Metody 13(1), 143–151 (1977)

    Google Scholar 

  5. Chudnovsky, M., Geelen, J., Cunningham, W.: An algorithm for packing non-zero A-paths in group-labelled graphs. Combinatorica 28(2), 145–161 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dial, R.: Algorithm 360: Shortest path forest with topological ordering. Commun. ACM 12, 632–633 (1969)

    Article  Google Scholar 

  7. Ford, L., Fulkerson, D.: Flows in Networds. Princeton University Press, Princeton (1962)

    Google Scholar 

  8. Frank, A.: On connectivity properties of Eulerian digraphs. Ann. Discrete Math. 41, 179–194 (1989)

    Article  Google Scholar 

  9. Ibaraki, T., Karzanov, A., Nagamochi, H.: A fast algorithm for finding a maximum free multiflow in an inner Eulerian network and some generalizations. Combinatorica 18(1), 61–83 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. Karger, D., Levine, M.: Finding maximum flows in undirected graphs seems easier than bipartite matching. In: STOC ’98: Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, pp. 69–78. ACM, New York (1998)

    Chapter  Google Scholar 

  11. Karzanov, A.: Combinatorial methods to solve cut-dependent problems on multiflows. In: Combinatorial Methods for Flow Problems, vol. 3, pp. 6–69. Inst. for System Studies, Moscow (1979)

    Google Scholar 

  12. Karzanov, A.: Fast algorithms for solving two known problems on undirected multicommodity flows. In: Combinatorial Methods for Flow Problems, vol. 3, pp. 96–103. Inst. for System Studies, Moscow (1979)

    Google Scholar 

  13. Lovász, L.: On some connectivity properties of Eulerian graphs. Acta Math. Akad. Sci. Hung. 28, 129–138 (1976)

    Article  MATH  Google Scholar 

  14. Lovász, L.: Matroid matching and some applications. J. Comb. Theory Ser. B 28, 208–236 (1980)

    Article  MATH  Google Scholar 

  15. Mader, W.: Über die maximalzahl kantendisjunkter a-wege. Arch. Math. (Basel) 30, 325–336 (1978)

    MATH  MathSciNet  Google Scholar 

  16. Mader, W.: Über die maximalzahl kantendisjunkter H-wege. Arch. Math. (Basel) 31, 382–402 (1978)

    MathSciNet  Google Scholar 

  17. Pap, G.: A constructive approach to matching and its generalizations. Ph.D. Thesis, Eötvös University, Department of Operations Research (2006)

  18. Pap, G.: Packing non-returning A-paths. Combinatorica 27(2), 247–251 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. Pap, G.: Packing non-returning A-paths algorithmically. Discrete Math. 308(8), 1472–1488 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Schrijver, A.: A short proof of Mader’s S-paths theorem. J. Comb. Theory Ser. B 82, 319–321 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  21. Schrijver, A.: Combinatorial Optimization. Springer, New York (2003)

    MATH  Google Scholar 

  22. Vazirani, V.: Approximation Algorithms. Springer, New York (2001)

    Google Scholar 

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Authors and Affiliations

  1. Dept. of Mechanics and Mathematics, Moscow State University, Leninskie Gory, 119991, Moscow, Russia

    Maxim A. Babenko

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  1. Maxim A. Babenko
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Correspondence to Maxim A. Babenko.

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Supported by RFBR grants 03-01-00475, 05-01-02803, and 06-01-00122.

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Babenko, M.A. A Fast Algorithm for the Path 2-Packing Problem. Theory Comput Syst 46, 59–79 (2010). https://doi.org/10.1007/s00224-008-9141-y

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  • DOI: https://doi.org/10.1007/s00224-008-9141-y

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