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Stronger bounds and faster algorithms for packing in generalized kernel systems

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Abstract

In this paper, we consider integral (and fractional) packing problems in generalized kernel systems with a mixed family (gksmf). This framework generalizes combinatorial packing problems of several structures as, for instance, st-flows, arborescences, kernel systems, branchings, convex branchings, and covers in bi-set systems. We provide an algorithm, which improves by a factor of 2 on the upperbound for the size of a packing in a gksmf, provided by Leston-Rey and Wakabayashi. This in turn implies the following upperbounds, where n is the number of vertices, m the number of arcs, and r the number of root-sets of a digraph: m for the size of a packing of st-paths; \(m-n+2\) for the size of a packing of spanning arborescences; \(m+r-1\) for the size of a packing of spanning branchings; \(m+r-1\) for the size of a packing of Fujishige’s convex branchings; m for the size of a packing of dijoins of a restricted form; and \(m+r-1\) for the size of a packing in Frank’s bi-set systems with the mixed intersection property. Next, we consider the framework of uncrossing gksmf. We describe an algorithm for computing a packing in such a framework with the same upperbound guarantee. The time complexity of this algorithm implies an improvement over the best time complexity bounds known for computing a packing of spanning arborescences of Gabow and Manu, for computing a packing of spanning branchings of Lee and Leston-Rey, and for computing a packing of covers in a bi-set system with the mixed intersection property of Bérczi and Frank.

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References

  1. Barahona, F.: Fractional packing of \(T\)-joins. SIAM J. Discrete Math. 17, 661–669 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bérczi, K., Frank, A.: Variations for Lovász submodular ideas. In: Grötschel, M., Katona, G.O.H. (eds.) Building Bridges Between Mathematics and Computer Science, Bolyai Society, Series: Mathematical Studies, 19, pp. 137–164. Springer, Berlin (2008)

  3. Bérczi, K., Frank, A.: Packing Arborescences. Technical report 2009-4, Egerváry Research Group

  4. Cook, W., Fonlupt, J., Schrijver, A.: An integer analogue of Carathéodory’s theorem. J. Comb. Theory Ser. B 40, 63–70 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  5. Edmonds, J.: Edge-disjoint branchings. In: Rustin B (ed) Combinatorial Algorithms, pp. 91–96. Academic Press, New York (1973)

  6. Feofiloff, P., Younger, D.H.: Directed cut transversal packing for source-sink connected graphs. Combinatorica 7, 255–263 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  7. Frank, A.: Kernel systems of directed graphs. Acta Sci. Math. (Szeged) 41(1–2), 63–76 (1979)

    MathSciNet  MATH  Google Scholar 

  8. Frank, A.: Increasing the rooted-connectivity of a digraph by one. Math. Program. 84, 565–576 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Frank, A.: Connections in combinatorial optimization. Oxford Lectures in Mathematics and its Applications, vol. 38. Oxford University Press, Oxford (2011)

  10. Frank, A.: Rooted \(k\)-connections in digraphs. Discrete Appl. Math. 157, 1242–1254 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fujishige, S.: A note on disjoint arborescences. Combinatorica 30, 247–252 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gabow, H.N., Manu, K.S.: Packing algorithms for arborescences (and spanning trees) in capacitated graphs. Math. Program. 82, 83–109 (1998)

    MathSciNet  MATH  Google Scholar 

  13. Gijswijt, D., Regts, G.: On the Caratheodory Rank of Polymatroid Bases. arXiv:1003.1079 (2010)

  14. Hao, J., Orlin, J.B.: A faster algorithm for finding the minimum cut in a directed graph. J. Algorithms 17(3), 424–446 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kamiyama, N., Katoh, N., Takizawa, A.: Arc-disjoint in-trees in directed graphs. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 518–526 (2008)

  16. Lee, O., Leston-Rey, M.: A faster algorithm for packing branchings in digraphs. Discrete Math. Appl. (2015). doi:10.1016/j.dam.2015.05.016

  17. Leston-Rey, M.: Integral packing of branchings in capacitated digraphs. J. Comb. Optim. (2014). doi:10.1007/s10878-014-9768-3

    MATH  Google Scholar 

  18. Leston-Rey, M.: Um Arcabouço Generalizado Para Empacotamento de Ramificacões e Outras Estruturas Combinatórias, in Portuguese. Ph.D. Thesis (2012)

  19. Leston-Rey, M., Wakabayashi, Y.: Packing in generalized kernel systems. Math. Program. Ser. A 149(1–2), 209–251 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lovász, L.: On two minimax theorems on graph theory. J. Comb. Theory Ser. B 21, 96–103 (1976)

    Article  MATH  Google Scholar 

  21. Matsuoka, Y.: Fractional packing in ideal clutters. Math. Program. 133(1–2), 159–169 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pevzner, P.A.: Branching packing in weighted graphs. Selected Topics in Discrete Mathematics, American Mathematical Society Translations Series 2, vol. 158, pp. 185–200. American Mathematical Society, Providence, Rhode Island (1994)

  23. de Pina, J.C., Soares, J.: A new bound for the Carathéodory rank of the bases of a matroid. J. Comb. Theory Ser. B 88, 323–327 (2003)

    Article  MATH  Google Scholar 

  24. Schrijver, A.: Min-max relations for directed graphs. In: Bachem, A., Grotschel, M., Korte, B. (eds.) Bonn Workshop on Combinatorial Optimization (Bonn, 1980). Ann. Discrete Math. 16, 261–280 (1982)

  25. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, vol. 3. Springer, Berlin (2003)

    MATH  Google Scholar 

  26. Szegő, L.: Note on covering intersecting set-systems by digraphs. Discrete Math. 234, 187–189 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Computing, University of Campinas, Campinas, Brazil

    Orlando Lee

  2. Independent researcher, São Paulo, Brazil

    Mario Leston-Rey

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  1. Mario Leston-Rey
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  2. Orlando Lee
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Correspondence to Mario Leston-Rey.

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This paper is dedicated to my son, Pedro Rey.

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Leston-Rey, M., Lee, O. Stronger bounds and faster algorithms for packing in generalized kernel systems. Math. Program. 159, 31–80 (2016). https://doi.org/10.1007/s10107-015-0948-4

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  • DOI: https://doi.org/10.1007/s10107-015-0948-4

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